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A baseball player has a batting average of $0.205$ . What is the probability that he has exactly $1$ hits in his next $7$ at bats? The probability is

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Find probabilities using the addition rule

Full solution

Q. A baseball player has a batting average of

0.205

. What is the probability that he has exactly

1

hits in his next

7

at bats? The probability is

Calculate probability formula: Calculate the probability of getting exactly $1$ hit using the binomial probability formula, which is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $\binom{n}{k}$ is the binomial coefficient.
Calculate binomial coefficient: Here, $n=7$ ( $7$ at bats), $k=1$ (exactly $1$ hit), and $p=0.205$ (batting average). First, calculate the binomial coefficient ( $7$ choose $1$ ) which is $7$ .
Calculate $p^k$ : Now, calculate $p^k$ which is $0.205^1 = 0.205$ .
Calculate $(1-p)^{(n-k)}$ : Next, calculate $(1-p)^{(n-k)}$ which is $(1-0.205)^{(7-1)} = 0.795^6$ .
Multiply all parts: Now, multiply all the parts together: $7 \times 0.205 \times 0.795^6$ .
Perform final calculation: Perform the calculation: $7 \times 0.205 \times 0.795^6 = 7 \times 0.205 \times 0.377149 = 0.539$ .

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