Question

1. a baseball player has 10 at bats in a series and has a 30% chance of getting a hit each time. let x = the number of hits the player gets. find:

a. p(x = 2)
b. p(x < 4)
c. p(x > 2)
d. p(x = 0)
e. p(x < 3)
f. p(x ≥ 6)

Explanation:

Step1: Identify the distribution

This is a binomial distribution problem. The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 10\) (number of trials), \(p=0.3\) (probability of success), and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Calculate \(P(X = 2)\)

For \(n = 10\), \(k = 2\), \(p = 0.3\), \(1-p=0.7\)

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Step3: Calculate \(P(X<4)\)

\(P(X<4)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)\)
For \(k = 0\):

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For \(k = 1\):

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For \(k = 3\):

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\(P(X<4)=0.0282 + 0.1211+0.2335 + 0.2668=0.6496\)

Step4: Calculate \(P(X>2)\)

\(P(X>2)=1 - P(X\leq2)=1-(P(X = 0)+P(X = 1)+P(X = 2))\)
\(P(X = 0)+P(X = 1)+P(X = 2)=0.0282+0.1211 + 0.2335=0.3828\)
\(P(X>2)=1 - 0.3828=0.6172\)

Step5: Calculate \(P(X = 0)\)

As calculated in Step3, \(P(X = 0)=(0.7)^{10}\approx0.0282\)

Step6: Calculate \(P(X<3)\)

\(P(X<3)=P(X = 0)+P(X = 1)+P(X = 2)=0.0282+0.1211+0.2335 = 0.3828\)

Step7: Calculate \(P(X\geq6)\)

\(P(X\geq6)=P(X = 6)+P(X = 7)+P(X = 8)+P(X = 9)+P(X = 10)\)
For \(k = 6\):

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For \(k = 7\):

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For \(k = 8\):

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For \(k = 9\):

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For \(k = 10\):

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\(P(X\geq6)=0.0369+0.0090 + 0.0015+0.0001+0.0000=0.0475\)

Answer:

A. \(P(X = 2)\approx0.2335\)
B. \(P(X<4)\approx0.6496\)
C. \(P(X>2)\approx0.6172\)
D. \(P(X = 0)\approx0.0282\)
E. \(P(X<3)\approx0.3828\)
F. \(P(X\geq6)\approx0.0475\)