Question

it is computed that when a basketball player shoots a free - throw, the odds in favor of his making it are 21 to 3. find the probability that when this basketball player shoots a free - throw, he misses it. out of every 100 free throws he attempts, on the average how many should he make? the probability that the player misses the free throw is (type an integer or a simplified fraction )

Explanation:

Step1: Recall the relationship between odds and probability

The odds in favor of an event $E$ is given by $\text{Odds}=\frac{P(E)}{1 - P(E)}$. Here, the odds in favor of making a free - throw is $\frac{21}{3}=7$. Let $P$ be the probability of making a free - throw. So, $\frac{P}{1 - P}=7$.

Step2: Solve the equation for $P$

Cross - multiply the equation $\frac{P}{1 - P}=7$ to get $P = 7(1 - P)$. Expand to $P=7 - 7P$. Add $7P$ to both sides: $P + 7P=7$, so $8P = 7$ and $P=\frac{7}{8}$.

Step3: Find the probability of missing the free - throw

The probability of missing the free - throw is $1 - P$. Substitute $P=\frac{7}{8}$ into it, we get $1-\frac{7}{8}=\frac{1}{8}$.

Step4: Find the number of made free - throws out of 100 attempts

If the probability of making a free - throw is $P=\frac{7}{8}$, and the number of attempts $n = 100$. The expected number of made free - throws is $n\times P$. So, $100\times\frac{7}{8}=\frac{700}{8}=87.5$.

Answer:

The probability that the player misses the free - throw is $\frac{1}{8}$.
On average, out of 100 free - throws, he should make 87.5.