• The “same-side bias” is alive and well in the simple act of the coin toss, with the side of the coin facing up to start more likely to be the side that lands up.
• Researchers say they’ve proven out an old Stanford theory by flipping coins over 350,000 times and landing on a trend.
• The 50.8 percent same-side bias features “strong evidence” and “compelling statistical support.”

Before you call heads or tails, peek at the side of the coin facing up. You’ll improve your odds of getting it right by calling for the side facing up. Scientists say the “overwhelming evidence for a same-side bias” is proved out by the results of 350,757 coin flips.

Frantisek Bartos, a psychological methods PhD candidate at the University of Amsterdam, led a pre-print study published on arXiv that built off the 2007 paper from Stanford mathematician Persi Diaconis asserting “that when people flip an ordinary coin, it tends to land on the same side it started.”

Diaconis estimated the same-side outcome was 51 percent. “Our data lend strong support to this precise prediction,” Bartos writes in the paper, “the coins landed on the same side more often than not.”

With a same-side bias in the new study played out across more than 350,000 coin flips at 50.8 percent, Bartos says that may not seem like powerful evidence, but he calls it “compelling statistical support for Diaconis’ physics model of coin tossing.”

In the 2007 paper, Diaconis says that “coin tossing is physics not random.” He points to how a spring-loaded coin tossing machine can be manipulated to ensure a coin starting heads-up lands that way 100 percent of the time. When the spring releases and the coin flips, it has a natural spin. Careful adjustment can work the physics into a precise landing point. “Naturally tossed coins obey the laws of mechanics and their flight is determined by their initial conditions,” he writes.

In a series of posts on X, formerly Twitter, Bartos says the value—and impressiveness—of the same-side bias gets spelled out when you turn to betting. “If you bet a dollar on the outcome of a coin toss 1,000 times, knowing the starting position of the coin toss would earn you $19 on average,” he writes. “This is more than the casino advantage for six-deck blackjack against an optimal player ($5), but less than that for single-zero roulette ($27).”

The study also tossed in an extra element and found no heads-tails bias, noting that when the initial side-up was randomly determined, the coin was equally likely to land heads or tails. “This lack of heads-tails bias,” the study says, “does not appear to vary across coins.”

For Diaconis in 2007, he was eager to “show that vigorously flipped coins tend to come up the same way they started.” Bartos believes he “overwhelmingly” proved out the old theory. And he did so with a flip (or 350,000) of a coin.