Probability - Flipping Coins

Flipping A Single Coin

On flipping a coin, you can get either heads or tails. Thus the sample space is {heads, tails} and total number of possible outcomes on flipping a coin is 2.

Suppose you want to find the probability of getting heads on flipping a coin. Since the coin can land on either heads or tails, hence there is 1 favorable outcome - heads. Applying the formula for theoretical probability,

`P("heads") = "Number of favorable outcomes"/"Total number of outcomes" = \frac{1}{2}`

Flipping Two Coins - or One Coin Twice

First, note that flipping two coins together or flipping just one coin twice - both experiments are same.

On flipping two coins, you can get a combination of heads and tails on the two coins. You can get heads on both coins, tails on both coins, heads on the first and tail on the second, and tails on the first and heads on the second. Thus, the sample space is {HH, HT, TH, TT}, where H represents heads and T represents tails.

From the sample space, there are a total of four possible outcomes.

Now suppose you have to find the probability of getting heads on the first coin and tails on the second coin. There is a single outcome in the sample space matching this. Thus there is 1 favorable outcome.

`P("HT") = "Number of favorable outcomes"/"Total number of outcomes" = \frac{1}{2}`

Flipping Three Coins - or One Coin Thrice

As mentioned above, flipping three coins simultaneously or flipping a single coin three times - both are the same experiments.

The possible number of combinations of heads and tails on three coins is naturally greater than those on two coins. The sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Thus there are 8 diffferent possible outcomes when three coins are flipped together.

Suppose you have to calculate the probability of getting three tails. There is only one favorable outcome matching this in the sample space. Thus,

`P("TTT") = "Number of favorable outcomes"/"Total number of outcomes" = \frac{1}{8}`

Finding probability in case of flipping a coin (or doing any other experiment) more than 1 times is easier and faster using the binomial theorem.