Question

1. patrick mahomes completes 70% of his pass attempts. assume he tries 15 passes in one quarter. let x = the number of passes completed. find: a. p(x = 10) b. p(x < 8) c. p(x > 14) d. p(x = 12) e. p(x < 13) f. p(x ≥ 12)

Explanation:

Step1: Identify the distribution

Since each pass attempt is a Bernoulli trial (success - pass completed, failure - pass not completed) with \(n = 15\) independent trials and probability of success \(p=0.7\), \(X\sim B(n = 15,p = 0.7)\). The probability mass function is \(P(X=k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\), where \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\)

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

For \(n = 15,k = 10,p=0.7,1-p = 0.3\)

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Step3: Calculate \(P(X<8)=\sum_{k = 0}^{7}P(X=k)\)

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Step4: Calculate \(P(X>14)=P(X = 15)\)

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Step5: Calculate \(P(X = 12)\)

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Step6: Calculate \(P(X<13)=\sum_{k=0}^{12}P(X = k)=1 - P(X\geq13)\)

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Step7: Calculate \(P(X\geq12)=P(X = 12)+P(X = 13)+P(X = 14)+P(X = 15)\)

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Answer:

A. \(P(X = 10)\approx0.206\)

B. \(P(X<8)\approx0.103\)

C. \(P(X>14)\approx0.0047\)

D. \(P(X = 12)\approx0.170\)

E. \(P(X<13)\approx0.874\)

F. \(P(X\geq12)\approx0.296\)