Probability the Double-Headed Coin Was Drawn After Ten Heads From a Jar of 1,000 Coins

A jar holds 1,000 coins: 999 are fair and one has heads on both sides. Coins are drawn from the jar at random and tossed, and ten heads come up. What is the probability that the double-headed coin was drawn? The report's wording is that ten are drawn and tossed and that all ten show heads. Read literally, that means ten different coins, each tossed once; the more familiar form of this puzzle draws a single coin and tosses it ten times. Agree on the experiment with the interviewer, and be ready to answer both readings: 1. Ten coins are drawn (assume without replacement) and each is tossed once; all ten land heads. What is the probability that the double-headed coin is among them? 2. One coin is drawn and tossed ten times; all ten tosses land heads. What is the probability that it is the double-headed coin? ```hint Two hypotheses Compare how likely ten heads are when the double-headed coin is involved and when it is not, and weight each case by how likely it was before any toss. ``` ```hint Count the fair tosses In each reading, ask how many of the ten heads are guaranteed if the double-headed coin was drawn. ``` ### Clarifying Questions - Is one coin tossed ten times, or are ten different coins each tossed once? - If ten coins are drawn, are they drawn without replacement? - Is every coin equally likely to be drawn? ### What a Strong Answer Covers - The correct prior probability of drawing the double-headed coin in each reading - The likelihood of ten heads under each hypothesis - Bayes' rule carried through to an exact fraction and a decimal value - An intuitive check, such as the odds form, that explains why the two readings give such different answers ### Follow-up Questions - In the one-coin reading, how many heads in a row are needed before the double-headed coin becomes more likely than not? - After ten heads in the one-coin reading, what is the probability that the next toss of the same coin is heads? - How do the answers change if the jar holds N fair coins and one double-headed coin?