Question

1. zach is an 85% field goal kicker. each goal he kicks is independent of the next kick. at practice today, he will be doing 30 kicks.

for the following table, fill in the missing pieces.

situation	probability notation	formula	value
what is the probability zach will make exactly 20 kicks?	$p(x = 20)$	$\dbinom{30}{20} 0.85^{20} 0.15^{10}$	0.0067
what is the probability zach will make no more than 20 kicks?	$p(x \leq 20)$		0.0097
what is the probability zach will make no more than 24 kicks?	$p(x \leq 24)$
	$p(x = 28)$	$\dbinom{30}{28} 0.85^{28} 0.15^2$
	$p(x \geq 27)$	$p(x = 27) + p(x = 28) + p(x = 29) + p(x = 30)$

Explanation:

Step1: Probability formula for binomial distribution

The binomial probability formula is \(P(X = k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success on a single trial. Here \(n = 30\), \(p=0.85\), \(1 - p = 0.15\)

Step2: For \(P(X\leq20)\)

\(P(X\leq20)=\sum_{k = 0}^{20}\binom{30}{k}0.85^{k}0.15^{30 - k}\). Using a binomial probability calculator or software (such as a TI - 84 Plus: binomcdf(30, 0.85, 20)), we get the value.

Step3: For \(P(X\leq24)\)

\(P(X\leq24)=\sum_{k = 0}^{24}\binom{30}{k}0.85^{k}0.15^{30 - k}\). Using a binomial probability calculator or software (such as binomcdf(30, 0.85, 24)), we find the value.

Step4: For \(P(X = 28)\)

\(\binom{30}{28}=\frac{30!}{28!(30 - 28)!}=\frac{30\times29}{2\times1}=435\). Then \(P(X = 28)=\binom{30}{28}0.85^{28}0.15^{2}=435\times0.85^{28}\times0.15^{2}\approx0.2603\)

Step5: For \(P(X\geq27)\)

\(P(X\geq27)=P(X = 27)+P(X = 28)+P(X = 29)+P(X = 30)\)

\(\binom{30}{27}=\frac{30!}{27!(30 - 27)!}=\frac{30\times29\times28}{3\times2\times1}=4060\)

\(P(X = 27)=\binom{30}{27}0.85^{27}0.15^{3}\)

\(\binom{30}{29}=\frac{30!}{29!(30 - 29)!}=30\)

\(P(X = 29)=\binom{30}{29}0.85^{29}0.15^{1}\)

\(\binom{30}{30}=1\)

\(P(X = 30)=\binom{30}{30}0.85^{30}0.15^{0}=0.85^{30}\approx0.0019\)

Using a binomial probability calculator or software (such as 1 - binomcdf(30, 0.85, 26)), we get \(P(X\geq27)\approx0.7054\)

Answer:

Situation	Probability Notation	Formula	Value
What is the probability Zach will make exactly 20 kicks?	\(P(X = 20)\)	\(\binom{30}{20}0.85^{20}0.15^{10}\)	\(0.0067\)
What is the probability Zach will make no more than 20 kicks?	\(P(X\leq20)\)	\(\sum_{k = 0}^{20}\binom{30}{k}0.85^{k}0.15^{30 - k}\)	\(0.0097\)
What is the probability Zach will make no more than 24 kicks?	\(P(X\leq24)\)	\(\sum_{k = 0}^{24}\binom{30}{k}0.85^{k}0.15^{30 - k}\)	\(\approx0.1662\)
What is the probability Zach will make exactly 28 kicks?	\(P(X = 28)\)	\(\binom{30}{28}0.85^{28}0.15^{2}\)	\(\approx0.2603\)
What is the probability Zach will make at least 27 kicks?	\(P(X\geq27)\)	\(P(X = 27)+P(X = 28)+P(X = 29)+P(X = 30)\)	\(\approx0.7054\)