Question

a tennis player makes a successful serve 70% of the time. assume that each serve is independent of the others. if she serves 6 times, what is the probability that she gets

1. all 6 serves in?

2nd, distr, binomialpdf ()=

1. exactly 4 serves in?

2nd, distr, binomialpdf ()=

1. at least 4 serves in?

1 - 2nd, distr, binomialcdf ()=

1. no more than 4 serves in?

2nd, distr, binomialcdf ()=

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 \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success on a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Solve problem 11

For \(n = 6\), \(k=6\), \(p = 0.7\)
\(C(6,6)=\frac{6!}{6!(6 - 6)!}=1\)
\(P(X = 6)=C(6,6)\times(0.7)^{6}\times(1 - 0.7)^{6-6}\)
\(P(X = 6)=1\times(0.7)^{6}\times(0.3)^{0}=(0.7)^{6}\approx0.117649\)

Step3: Solve problem 12

For \(n = 6\), \(k = 4\), \(p=0.7\)
\(C(6,4)=\frac{6!}{4!(6 - 4)!}=\frac{6\times5}{2\times1}=15\)
\(P(X = 4)=C(6,4)\times(0.7)^{4}\times(0.3)^{2}\)
\(P(X = 4)=15\times0.2401\times0.09 = 15\times0.021609=0.324135\)

Step4: Solve problem 13

\(P(X\geq4)=P(X = 4)+P(X = 5)+P(X = 6)\)
For \(k = 5\), \(C(6,5)=\frac{6!}{5!(6 - 5)!}=6\)
\(P(X = 5)=C(6,5)\times(0.7)^{5}\times(0.3)^{1}=6\times0.16807\times0.3 = 0.302526\)
We already know \(P(X = 4)=0.324135\) and \(P(X = 6)\approx0.117649\)
\(P(X\geq4)=0.324135 + 0.302526+0.117649\approx0.74431\)

Step5: Solve problem 14

\(P(X\leq4)=1-(P(X = 5)+P(X = 6))\)
\(P(X = 5)+P(X = 6)=0.302526+0.117649 = 0.420175\)
\(P(X\leq4)=1 - 0.420175=0.579825\)

Answer:

1. \(\approx0.117649\)
2. \(0.324135\)
3. \(\approx0.74431\)
4. \(0.579825\)