Question

a hockey player is practicing shooting at the goal from the blue line. if the probability of his hitting the goal is 0.75, what is the probability he will hit the goal in at least 10 of his next 15 shoots? 0.6849 0.4227 0.8516 0.2205

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 15\) (number of trials), \(p=0.75\) (probability of success), \(1-p = 0.25\) (probability of failure), and \(C(n,k)=\frac{n!}{k!(n - k)!}\). We need to find \(P(X\geq10)=P(X = 10)+P(X = 11)+P(X = 12)+P(X = 13)+P(X = 14)+P(X = 15)\)

Step2: Calculate \(P(X = k)\) for \(k = 10\)

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Step3: Calculate \(P(X = k)\) for \(k = 11\)

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Step4: Calculate \(P(X = k)\) for \(k = 12\)

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Step5: Calculate \(P(X = k)\) for \(k = 13\)

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Step6: Calculate \(P(X = k)\) for \(k = 14\)

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Step7: Calculate \(P(X = k)\) for \(k = 15\)

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Step8: Sum up the probabilities

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(Using a binomial probability calculator or software (e.g., in Excel: BINOM.DIST.RANGE(15,0.75,10,15)), we get a more accurate value)

Using a binomial probability calculator:

$$P(X\geq10)=\sum_{k = 10}^{15}C(15,k)\times(0.75)^{k}\times(0.25)^{15 - k}\approx0.8516$$

Answer:

0.8516