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Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to randomly choose between two alternatives. It is a form of sortition which inherently has two possible outcomes.
The probability of 20 heads, then 1 head is 0.5 20 × 0.5 = 0.5 21; The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152. When flipping a fair coin 21 times, the outcome is equally likely to be 21 heads as 20 heads and then 1 tail.
A test is performed by tossing the coin N times and noting the observed numbers of heads, h, and tails, t. The symbols H and T represent more generalised variables expressing the numbers of heads and tails respectively that might have been observed in the experiment. Thus N = H + T = h + t.
Using for heads and for tails, the sample space of a coin is defined as: Ω = { H , T } {\displaystyle \Omega =\{H,T\}} The event space for a coin includes all sets of outcomes from the sample space which can be assigned a probability, which is the full power set 2 Ω {\displaystyle 2^{\Omega }} .
Flip the coin twice. If both comes up heads or tails, end the experiment. Else, flip the coin 4 more times. This experiment has 7 types of outcomes: 2 heads, 2 tails, 5 heads 1 tail, ..., 1 head 5 tails. We now calculate the p-value of the "3 heads 3 tails" outcome.
In the heads scenario, Sleeping Beauty would spend her wager amount one time, and receive 1 money for being correct. In the tails scenario, she would spend her wager amount twice, and receive nothing. Her expected value is therefore to gain 0.5 but also lose 1.5 times her wager, thus she should break even if her wager is 1/3.
A representation of the possible outcomes of flipping a fair coin four times in terms of the number of heads. As can be seen, the probability of getting exactly two heads in four flips is 6/16 = 3/8, which matches the calculations. For this experiment, let a heads be defined as a success and a tails as a failure.
Spinner spins a pair of heads before a pair of tails or odding out. Single Tail 3.125% 1–1 Spinner spins a pair of tails before a pair of heads or odding out. 5 Odds 9.375% 28–1 Spinner spins five odds in a row ("odding out") before either a pair of heads or a pair of tails. Spinner's Bet 3.400% 15–2 Only available to the current spinner.