Breaking the Law of Averages
By Jill Storey
I was always a sucker for the clarity of math, especially statistics. I loved the way statistics brought order to the world, sweeping the messiness of ignorance and chance into a neat pile of percentages. Because of statistics, I was never afraid during thunderstorms. The odds of being struck by lightning are one in 600,000 every year, which, it seemed to me, was virtually a free pass to sit on the porch during a storm and watch fissures of lightning rip through the heavens.
When I took a statistics course in graduate school, probability formulas drew me in with their soothing logic. People and events became numbers, and when I plugged those numbers into formulas, I could make sense of problems that had seemed shapeless and shadowy. Probability theory was a kind of faith: It gave answers to why things happened and a set of commandments for making choices. Luck and fate had no place in my belief system.
It wasn’t until much later that I understood the answer a famous physicist gave when asked why he hung a horseshoe over his front door. “I have been reliably informed,” he answered, “that it will bring me luck whether I believe in it or not.”
* * *
The atmosphere of the prenatal ultrasound lab at the university hospital was as clinical as its name. No soft-focus posters of smiling babies or of young mothers gazing soulfully at their half-moon bellies here—just the hard plastic and bright steel of machines and monitors, the toneless hum of fluorescent lights.
But I didn’t mind the sterile room, or even the standard-issue thin cotton gown I was wearing, through which I could feel the cold vinyl of the examining table along my back. I was eight weeks pregnant and extremely happy about it. I had been married earlier that year at age thirty-six, and my husband, Richard, and I were eager to start a family. I had read the medical literature and learned that the odds of my getting pregnant quickly were slim. According to the statistics, less than fifty percent of women my age become pregnant within the first year of trying. But within two months I was clutching a pregnancy test, an oracle in a white plastic stick with a glowing pink line that seemed to shout Yes! You’ve won! Because of my age, however, the baby had a higher risk of genetic disorders, so I was here for an ultrasound to determine what type of prenatal testing I needed.
The ultrasound technician spread cold gel on my stomach to smooth the movement of the transducer wand over my skin. Richard and I watched the images on the screen next to us as she moved the wand back and forth, but the blurry white smudges against the background of a black triangle were as inscrutable as the Milky Way.
“Did you take fertility drugs?” asked the technician. She was probably asking because of my age, I thought. When I said no, she probed further. “Clomid, assisted insemination, anything?” Again I told her no. She asked when I conceived, and when I told her the date, she seemed to distrust my answer. I could practically set my watch by my menstrual cycle, I said. She continued circling the wand over me. My husband gripped my hand.
“I’m going to have the doctor take a look at these pictures,” said the technician. She was gone for five long minutes. My husband and I whispered to each other. Was I having twins? Why didn’t she just say so? Why wasn’t she clear on my conception date?—it was right there on my chart.
The doctor came in and sat down, with that serious half-smile doctors wear in an attempt to soften bad news. “Sherry says you didn’t have any fertility treatments?” he asked. I thought the sound of my heart thumping would set off the ultrasound monitor. “I didn’t,” I replied. “What’s wrong?”
He pointed to the image on the screen. “You have three fetuses.” Now I could see three wavery little beans on the monitor.
“You mean I’m having triplets?” I asked. My husband looked stunned.
“Well, not anymore,” replied the doctor. “I’m afraid you’ll be lucky if one survives. One has a good heartbeat. But it’s only the size of a typical six-week-old fetus. One is even smaller and has a weak heartbeat. One, I’m sorry to say, has no heartbeat at all.”
I tried to make sense of this information. Was I still going to have a baby? How many? Does one fetus with no heartbeat get subtracted from two live ones? The arithmetic overwhelmed me. I looked at Richard for guidance. How is it possible for grief and hope to coexist in the same equation?
Richard finally spoke. “So what do we do now?”
“Come in next week, and we’ll check to see if those two are growing,” answered the doctor. He paused. “I’ve never seen a triplet pregnancy without fertility drugs. They’re very rare.”
Rarer, it turns out, than a lightning strike. Natural triplets—those conceived without fertility drugs—occur in only one of every eight thousand pregnancies. Since the odds of being hit by lightning are one in 600,000 every year, if I live to be eighty-five years old, my lifetime odds would be about one in seven thousand. Better than the odds of having the hearts of three babies beating a tiny riff in my womb.
One week later, I lay on the same exam table. This time, the doctor did the ultrasound himself. I didn’t watch the monitor as the wand danced its clumsy waltz over my belly. I couldn’t bear to see them if I couldn’t keep them.
The doctor put his hand on my arm. “I’m sorry,” he said gently. “There are no heartbeats.”
I nodded mutely.
“I’ll call your OB/GYN. You can wait to miscarry, or Dr. Lee can do a D&C.” He left Richard and me to adjust to our new future, wiped clean of anticipations: no happy announcement, no due date, no baby. Times three.
I hated the thought of the little bodies inside me, entombed in my uterus. We went straight to Dr. Lee’s office. The procedure was quick and efficient—”Here come the cramps; count to ten; breathe; you did great”—and I went home to ponder the question, why me?
* * *
The ancient Greeks believed that the gods controlled everything: birth and death, earthquakes and droughts, the color of your child’s eyes, the deer that crossed your path three days in a row. For more than two thousand years, across continents, people continued to believe that whatever happened was part of a divine plan. Whether their gods lived on Mount Olympus or ruled in heaven or appeared in the form of an eagle, people took comfort in the idea that there was a reason for everything.
Then came the Renaissance, which encouraged scientific inquiry and the questioning of old ways of thinking. One of the questioners was an Italian mathematician and gambler named Girolamo Cardano, who discovered the laws of probability. I picture him at his oak desk at the University of Padua, two ivory dice clenched in his hand. Over and over and over again, he rolls the dice across the desktop, tallying how often each number comes up. I wonder if the sound of ivory clattering over the wood began to irritate him, and if he spread his handkerchief down to soften the noise. Perhaps his hand began to cramp, and he had to switch to his left hand. How long did it take—days, weeks?—to discover that lucky sevens came up not when he blew on the dice, or said a prayer, but, on average, one out of six times? Did Cardano—weary from lack of sleep, black dots dancing in front of his eyes even when he rested them briefly— swear off gambling after he learned that the probability of throwing snake eyes or double sixes was 2.8%, and no promises to heaven would ever change that?
Or perhaps my image of Cardano is all wrong. Maybe he simply worked out the calculations on his slate one warm summer afternoon when no students came seeking his counsel. If so, I doubt he stopped gambling. It’s easier to ignore mathematical theory than the reality of a cramped hand.
* * *
Six months later, I was pregnant again. Like a gambler switching blackjack tables after a big loss, I switched to a new doctor, a new chance. We didn’t tell anyone about the new pregnancy; we hardly talked about it ourselves. Unlike the first time, I didn’t dare get excited. I didn’t wonder if it was a boy (fifty-one percent chance) or a girl (forty-nine percent). I just counted the days until my ultrasound appointment.
That day never came. Six weeks into the pregnancy, while sitting in my office developing a spreadsheet, I started bleeding. I called Dr. Raskin’s office in a panic. “Go home and lie down,” the nurse advised me. “There’s nothing we can do.”
This time, I didn’t need a D&C. My uterus was already emptying itself, cramp by painful cramp. I was upset, of course, but it was easier this time, since I had never seen the peanut-shaped ghost on the ultrasound monitor. The embryo’s wisp of a heart never beat through the technician’s wand. This was just average bad luck, I tried to tell myself. After all, one in five pregnancies ends in miscarriage. The cramping stopped after a couple of days, and I went back to work.
A few days later I was absorbed in my spreadsheet, tinkering with the sales projections, when I felt a cramp. A pause, and then another. Damn, I thought, I didn’t bring any painkillers. I tried to work, but the cramps became sharper until I could no longer concentrate. I figured I’d better call Dr. Raskin, maybe get a prescription painkiller. When I got through to her office, she told me to come in right away. “You’re not supposed to be cramping anymore.”
By the time Richard and I arrived at her office, sharp waves of pain gripped my pelvis. The pain was so bad I was afraid to tell the doctor about it; I had the irrational thought that if I didn’t admit the intensity of the pain, it would stop, and I could go home and return to having an average miscarriage.
Dr. Raskin immediately ushered us into her office. She tried to palpate my abdomen gently, but I gasped at each probe of her fingertips. Richard watched helplessly.
“What do you think it might be?” he asked.
“It could be that you didn’t actually miscarry,” she replied. “Cramping at this stage is consistent with a tubal pregnancy, but your fallopian tubes feel normal.” I knew something about tubal pregnancies, where the embryo, on its way from the ovary to the uterus, implants in the fallopian tube instead of completing the journey. There the embryo grows, stretching the tube until it bursts, ending the pregnancy and destroying the tube. One to two percent of all pregnancies are tubal.
There is a story about a statistics professor who lived in Moscow during World War II. When the German planes strafed the city, the professor never rushed to the air-raid shelter with the rest of his neighbors. “There are seven million people in Moscow,” he explained. “What are the odds that a bomb will hit me?” One evening, his neighbors were surprised to see the professor huddled in the shelter with them. “What made you change your mind?” they asked. “There are seven million people and one elephant in Moscow,” the professor replied. “Last night, they got the elephant.”
But I couldn’t be the elephant. Not me, not this time.
After all, Dr. Raskin said my tubes didn’t seem swollen.
“I’m going to send you across the street to ultrasound. We need to figure out what’s going on. Do you think you can walk there, or shall I find a wheelchair?”
“I can walk,” I said, still trying to mask my pain and fear. Richard and I took the elevator down and crossed the busy street.
“Do you want to rest?” he asked.
“No, I just want to get there quickly.” When we entered the lobby we saw that only one elevator was working and dozens of people were waiting. “Let’s take the stairs,” I said. My panic somehow propelled me up four flights of stairs. As we entered the ultrasound waiting room I turned to Richard and said, “I think I’m going to pass out.”
The next thing I knew I was lying on a table in an ultrasound room. Richard and a woman in a white coat hovered over me. I was exhausted, light-headed, and peacefully, blissfully pain-free. “What happened?” I asked.
“I’m Dr. Sorenson,” the woman replied. “You’re probably in shock; don’t sit up.” She didn’t need to warn me; all I wanted was to lie there, luxuriating in the quiet, calm absence of pain.
Again the cold gel and the wand moving back and forth as if my abdomen were a Ouija board. Dr. Sorenson watched the screen closely. “There’s a lot of fluid in your peritoneal cavity,” she said.
“Fluid?” I pictured my womb floating on gentle waves.
“Possibly blood. We don’t know. I’m going to call your doctor.”
Dr. Sorenson returned a few minutes later. “We don’t know if this is a tubal pregnancy, and the only way to find out is to look inside. Dr. Raskin is waiting for an operating room to become available.”
An orderly transferred me onto a gurney and wheeled me into the emergency room to wait. The gurneys were lined up in two neat rows, each bed holding a patient facing a sudden crisis. Next to me, a heavyset woman rolled her head back and forth, muttering, “Help me Jesus! Oh, Jesus, Jesus, please help me!” Across the aisle a man held a woman’s limp hand and explained to her, “You OD’ed, baby. I found you on the couch.”
While Richard left to call my family, a nurse sat down next to me and asked how I was feeling.
Physically, I was only sore, but I was exhausted and frightened. “I’m scared,” I said.
“Tell me what you’re afraid of,” she said gently.
“I’m afraid I’m hemorrhaging or something.”
“Don’t worry, we’re monitoring you, and you’re doing fine. Is there anything else?”
“Yes,” I whispered. “I’m afraid I’m going to lose my tube and I’ll never have a baby.”
“Women can get pregnant with one tube,” she said. “We’re going to take care of you.” The noise in the room seemed to recede. I closed my eyes and tried to believe her.
* * *
After the surgery, after I stopped throwing up from the anesthesia, after they let me wet my parched throat with ice chips, Dr. Raskin came in and sat next to me. “It wasn’t a tubal pregnancy, Jill,” she said. “It was similar, though. The embryo implanted in your ovary, and your ovary ruptured. That’s what made you pass out. The fluid they saw on the ultrasound was blood from the rupture. I saved some of the ovary, but it was pretty messy.”
I tried, in my woozy state, to absorb this news. “My ovary exploded?” I imagined my ovary, a small bright planet in the darkness of my abdomen, bursting open in a shower of fireworks.
“Well, basically, yes. It’s called an ovarian pregnancy. I’ve read about them, but I’ve never had a patient with one. It’s so lucky you were at the hospital when it ruptured. Otherwise, you could have died, you know.” She patted my hand. “I’ll let you rest now.”
She left the room and gently closed the door. I turned my head into the scratchy pillow and, for the first time all day, I cried.
* * *
There is a concept in probability theory called “independence.” It means that in any set of random occurrences, such as a set of ten coin tosses, each occurrence is completely independent from those that came before it. In other words, if you toss a coin nine times and it comes up heads each time, the probability of it coming up tails on the tenth throw are still fifty percent, no more, no less. The outcome of the tenth throw has nothing to do with the outcome of the previous nine. So now my pregnancies comprised a set of two random and independent occurrences.
My triplet pregnancy was one in eight thousand, and my ovarian pregnancy was, I learned, one in ten thousand. According to probability theory and the doctors, these anomalies were completely independent of each other. I know the ancient Greeks never would have believed that. But did I How could not just one but two such improbable things have happened to me.
In biblical times, communities sometimes identified guilty people by the drawing of lots, believing that God would cause the lot of the criminal to be drawn. Rabbi Isaac ben Mosheh Aramah, a fifteenth-century scholar, was more cautious: He thought that the casting of a single lot was a chance occurrence, but if that same lot was drawn repeatedly, it was a sign from God that the person was guilty. My lot had been drawn twice, from thousands and thousands of lots. What did it mean? It was easy to believe in random chance when other people suffered misfortunes. When my turn came, though, I felt cursed.
* * *
Two days after the operation, Dr. Raskin came into my hospital room. She sat down next to the bed and sighed. “Bad news, I’m afraid. Your blood tests show that we didn’t succeed in removing all of the fetal tissue, so it’s continuing to grow. But since the pregnancy was outside your uterus, a D&C won’t work.”
It sounded like science fiction. Pieces of a baby gripping my insides, trying to regenerate. “So what do we do?” I asked. “I’ve consulted with some of the specialists here, and we’re going to try an experimental drug, methotrexate. It’s used in treating cancer, but studies have shown success in using it to abort pregnancies. It should kill off any remaining tissue.”
So my pregnancy was a cancer, my body turning against me. I received the drug that afternoon, and, a couple of days later, my pregnancy finally let go.
When I left the hospital, Dr. Raskin told me that I needed to treat my body as if I had broken my pelvis. No one knew how long my recovery would be or what my chances were now of getting pregnant.
* * *
If I were living in Greco-Roman times, I would consult astragali to find out if there was a child in my future. Astragali, which were made from the knucklebones of animals, were similar to dice, except with four numbered sides instead of six. The person seeking to know what the fates held in store or to make the correct decision would pose a question and then throw four or five astragali. Each possible combination of numbers corresponded to an answer from the gods. Reading through the translations of the meanings of various throws, I came upon one interpretation of a throw that made me feel as if the gods were reaching through millennia to tell me something:
Three fours and two sixes. God speaks as follows:
Abide in thy house, nor go elsewhere,
Lest a ravening and destroying beast come nigh thee.
For I see not that this business is safe. But bide thy time.
The name of this combination of astragali was The throw of child-eating Cronos.
* * *
I followed my doctor’s advice, took a leave of absence from my job and abided in my house. But over the next days and months, my body continued to betray me. First an infection landed me back in the hospital on IV antibiotics. Then, about a week after I came home, a sharp pain in my pelvic region took my breath away. Five minutes later, another one. The pains were intermittent, like knife stabs, so intense it made the pain from my ovarian pregnancy seem like mere menstrual cramps. Richard took me to the emergency room, where they examined me with the ultrasound machine. The exam found nothing amiss; the “ravening and destroying beast” did not appear on the screen. Maybe it was a symptom of scar tissue forming from my surgery, said the doctor. A shot of painkiller and I was sent home. The following week the pains came again. Another trip to the emergency room, another ultrasound, more baffled doctors. They sent me home with powerful painkillers that made me vomit uncontrollably. The episodes recurred several times. I was afraid to go out, afraid to be left alone. “It’s just pain,” shrugged one doctor, dismissing my fears.
There was no name for whatever was attacking my insides. There was nothing I could look up, no facts, no variables this time to calculate the odds of these episodes occurring. I was weak from the surgery, the recurrent and unpredictable pains, the emotional toll of repeated loss.
Six weeks later, the attacks ended as mysteriously as they had begun. The doctors cleared me to try to conceive again. Months of hope punctured by disappointment passed. I had more surgery to clean out the scar tissue from the rupture the doctors thought might be preventing pregnancy. By now I was thirty-eight, and my doctor thought I should try fertility treatments.
The fertility specialist recommended I start with a drug called Pergonal, delivered in a series of injections that stimulate egg production. He told me the statistics: Women my age have about a one in twelve chance of getting pregnant with the drug. With my medical history, probably less, and the process would be expensive and time-consuming.
The statistics were discouraging. Would it be worth trying? For thousands of years the Chinese used the I Ching as an oracle. The questioner cast stalks or coins or dice and matched the pattern with one of the sixty-four hexagrams. The I Ching’s explanation of the hexagram foretold the person’s fate. That changed, according to scholar Richard Wilhelm, the first time a person, instead of passively accepting his fate, asked what he should do. By taking his “fate” into his own hands, he turned the book of divination into a book of wisdom.
Maybe I could change my fate. I went for the fertility drugs.
I conceived the first month of treatment. At six weeks, I again offered my belly to the ultrasound machine. I again watched the black-and-white screen, that modern crystal ball that had so often been the bearer of bad tidings. In the swirling snow on the screen, a shape coalesced: a constellation of white dots with a quivering star at the center. One fetus, one strong heartbeat.
* * *
During my pregnancy, I knew there was still much that could go wrong. I could miscarry (ten percent probability after confirming a heartbeat); the fetus could have a chromosomal defect (one percent chance at my age) or be afflicted by some random disease. But I realized that I couldn’t control or even predict what was going to happen by paying attention to probabilities. Pregnancy statistics, like all statistics, rely on samples, a group chosen to represent the whole. A sample, then, is simply a metaphor; it suggests a larger truth, but it is not the truth itself. I could interpret the metaphor in my own way.
My faith in probabilities hadn’t protected me from ending up as an outlier; luck, it seemed, didn’t care what I believed in. But what did I believe now? I thought about how runs of good or bad luck seem to inevitably run out, just as, in laws of statistics, everything regresses towards the mean. Maybe luck and fate obey laws of the universe that we simply haven’t discovered yet.
I decided then to let go of the statistics. In past ages, even the most brilliant mathematicians didn’t dare calculate probabilities, because to do so meant challenging the mysteries of the divine.
* * *
I gave birth to a baby girl on a rainy November night. There were no stars out to witness her arrival; there may have been lightning in the far hills. The nurses wrote down her vital statistics; she was, to everyone but us, an average newborn.
Twelve years have passed now. Our daughter is very good with numbers. She is also extremely lucky at games of chance.
Author’s Note: I am still fascinated by both the science and the psychology of probability. Most people are overly concerned about low probability events (e.g., plane crashes) yet nonchalant about higher probability events (e.g., car crashes). As a math-minded person, this never made sense to me, but now I understand it better. When you’re the one out of ten thousand—whether it’s having a rare disease or winning the lottery—it feels as if you’ve been singled out, separated from the rest of the world. If no one shares your experience, your suffering seems more intense and your joy more special. For me, the lightning bolts I endured in my quest to have a baby were blunted by that most ordinary of joys, the birth of my daughter, and, two years later, the birth of my son.
Jill Storey lives in San Francisco with her husband and two children. Her articles and essays have been published in Salon, the Washington Post, the San Francisco Chronicle, BabyCenter, Ms. Magazine and other publications.
Brain, Child (Fall 2008)