Use a binomial distribution to calculate probability
Determine if a probability model meets the conditions for a binomial distribution
Binomial Distribution
If the event [latex]A \text{ and } B[/latex] are mutually exclusive (meaning they cannot happen at the same time), then [latex]P(A~or~B) =P(A)+P(B).[/latex]
Let’s revisit the experiment flipping the coin [latex]3[/latex] times and counting the number of tails obtained.
Find [latex]P(X = 1).[/latex]
Since the outcomes THH, HTH, and HHT are mutually exclusive, the probability of obtaining [latex]1[/latex] tail in [latex]3[/latex] coin flips, i.e., [latex]P(X = 1)[/latex] , is:
where, as you observed in the table below, [latex]3[/latex] is the number of ways of obtaining [latex]1[/latex] tail in [latex]3[/latex] coin flips.
Recall that the outcomes of the experiment are as given in the following table:
Experimental Outcome
[latex]X[/latex]
Number of Tails in 3 Flips of a Coin
HHH
[latex]0[/latex]
HHT
[latex]1[/latex]
HTH
[latex]1[/latex]
THH
[latex]1[/latex]
TTH
[latex]2[/latex]
THT
[latex]2[/latex]
HTT
[latex]2[/latex]
TTT
[latex]3[/latex]
The example above gives us a glimpse into the formula for the binomial distribution, which is used to model a binomial experiment.
binomial distribution formula
In general, the formula for the probability of obtaining [latex]x[/latex] successes from [latex]n[/latex] independent trials where the probability of success is [latex]p[/latex] is:
[latex]P(X = x) = (\text{number of ways to obtain }x\text{ successes in }n\text{ trials}) \cdot p^{x} \cdot (1-p)^{n-x}[/latex]
where [latex]p^{x}[/latex] occurs because there are [latex]x[/latex] successes, and [latex](1-p)^{n-x}[/latex] occurs because if there are [latex]x[/latex] successes and [latex]n[/latex] trials total, there must be [latex]n-x[/latex] failures.