Question

1. if e and f are dependent events, then which of the following must be true?

e. ( p(ecap f)=p(e)p(f) )
f. ( p(ecap f)=p(e)p(f|e) )
g. ( p(ecap f)=p(e)+p(f) )
h. ( p(ecap f)=0 )

1. the table of data shows hit information for four well - known baseball players. suppose that one hit from the table is randomly selected. find each of the following probabilities. write your answer as a fraction in the simplest form.

a. find ( p(dj) ). (probability that a hit was made by derek jeter.)
b. find ( p(hr) ). (probability that a hit was a home run.)
c. find ( p(dj ) and ( hr) ). (probability that a hit was made by derek jeter and it was a home run)
d. find ( p(dj ) or ( hr) ). (probability that a hit was made by derek jeter or it was a home run.)
e. find ( p(rc ) or ( is) ). (probability that a hit was made by roberto clemente or ichiro suzuki.)
f. find ( p(hr|dj) ). (probability that a hit was a home run given that the hit was made by derek jeter.)
g. find ( p(pr|s) ). (probability that a hit was made by pete rose given that it was a single.)

Explanation:

Step1: Recall the probability formula

The probability formula is \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of elements in event \(A\) and \(n(S)\) is the number of elements in the sample space.

Step2: Calculate \(P(DJ)\)

The total number of hits \(n(S) = 17142\). The number of hits by Derek Jeter \(n(DJ)=4335\). So \(P(DJ)=\frac{4335}{17142}=\frac{4335\div3}{17142\div3}=\frac{1445}{5714}\)

Step3: Calculate \(P(HR)\)

The number of home - runs \(n(HR)=260 + 240+117 + 777=1394\). So \(P(HR)=\frac{1394}{17142}=\frac{1394\div2}{17142\div2}=\frac{697}{8571}\)

Step4: Calculate \(P(DJ\cap HR)\)

The number of home - runs by Derek Jeter \(n(DJ\cap HR) = 260\). So \(P(DJ\cap HR)=\frac{260}{17142}=\frac{260\div2}{17142\div2}=\frac{130}{8571}\)

Step5: Calculate \(P(DJ\cup HR)\)

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Here \(A = DJ\) and \(B = HR\). \(P(DJ)=\frac{4335}{17142}\), \(P(HR)=\frac{1394}{17142}\), \(P(DJ\cap HR)=\frac{260}{17142}\). Then \(P(DJ\cup HR)=\frac{4335 + 1394-260}{17142}=\frac{5469}{17142}=\frac{5469\div3}{17142\div3}=\frac{1823}{5714}\)

Step6: Calculate \(P(RC\cup IS)\)

The number of hits by Roberto Clemente \(n(RC) = 3846\), the number of hits by Ichiro Suzuki \(n(IS)=3664\). \(P(RC\cup IS)=\frac{3846+3664}{17142}=\frac{7510}{17142}=\frac{7510\div2}{17142\div2}=\frac{3755}{8571}\)

Step7: Calculate \(P(HR|DJ)\)

Use the formula \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). Here \(A = DJ\) and \(B = HR\). \(P(HR|DJ)=\frac{\frac{260}{17142}}{\frac{4335}{17142}}=\frac{260}{4335}=\frac{260\div5}{4335\div5}=\frac{52}{867}\)

Step8: Calculate \(P(PR|S)\)

The number of singles \(n(S)=13810\). The number of singles by Pete Rose \(n(PR\cap S)=4256\). Use the formula \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). Here \(A = S\) and \(B = PR\). \(P(PR|S)=\frac{\frac{4256}{17142}}{\frac{13810}{17142}}=\frac{4256}{13810}=\frac{4256\div2}{13810\div2}=\frac{2128}{6905}\)

Answer:

a. \(\frac{1445}{5714}\)
b. \(\frac{697}{8571}\)
c. \(\frac{130}{8571}\)
d. \(\frac{1823}{5714}\)
e. \(\frac{3755}{8571}\)
f. \(\frac{52}{867}\)
g. \(\frac{2128}{6905}\)