Skill: Calculate binomial probabilities

A binomial probability is used when:

• There is a fixed number of trials, $n$.
• Each trial has only two outcomes: success or failure.
• The trials are independent.
• The probability of success, $p$, stays the same on every trial.

The probability of getting exactly $k$ successes is

$$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k},$$

where:

• $n$ = total number of trials
• $k$ = number of successes wanted
• $p$ = probability of success
• $1-p$ = probability of failure
• $\binom{n}{k}$ = number of ways to arrange the $k$ successes among the $n$ trials

Example

A basketball player makes a free throw with probability $0.7$. What is the probability that the player makes exactly $3$ out of $5$ free throws?

Step 1: Identify the values

There are $5$ trials, so $n=5$.

The desired number of successes is $3$, so $k=3$.

The probability of success is $p=0.7$.

The probability of failure is

$$1-p=1-0.7=0.3.$$

Step 2: Substitute into the formula

$$P(X=3)=\binom{5}{3}(0.7)^3(0.3)^{5-3}$$

Step 3: Calculate each part

$$\binom{5}{3}=10$$

So,

$$P(X=3)=10(0.7)^3(0.3)^2$$ $$P(X=3)=10(0.343)(0.09)$$

Step 4: Find the probability

$$P(X=3)=0.3087$$

Therefore, the probability that the player makes exactly $3$ out of $5$ free throws is

$$\boxed{0.3087}$$

or approximately $\boxed{30.87\%}$.