Compute the following binomial probabilities directly from the formula for b(x, n, p):a) b(3, 8, .6)b) b(5, 8, .6)c) P(3 ≤ X ≤ 5) when n = 8 and p = .6d)P(1 ≤ X) when n = 12 and p = .1

Compute the following binomial probabilities directly from the formula for b(x, n, p):a) b(3, 8, .6)b) b(5, 8, .6)c) P(3 ≤ X ≤ 5) when n = 8 and p = .6d)P(1 ≤ X) when n = 12 and p = .1

Question
Decimals
asked 2020-10-20

Compute the following binomial probabilities directly from the formula for \(b(x, n, p)\):

a) \(b(3,\ 8,\ 0.6)\)

b) \(b(5,\ 8,\ 0.6)\)

c) \(\displaystyle{P}{\left({3}≤{X}≤{5}\right)}\)

when \(n = 8\) and \(p = 0.6\)

d)\(\displaystyle{P}{\left({1}≤{X}\right)}\) when \(n = 12\) and \(p = 0.1\)

Answers (1)

2020-10-21

Step 1

Solution

a) Given:\(\displaystyle{n}={8},{p}={0.6},{x}={3}\)

\(X \sim\) Binomeal \(\displaystyle{\left({x}={8},{p}={0.6}\right)}\) I will use the excel formula "=BINOM.DIST(x, n, p, FALSE)" to find the probability. \(\begin{array}{|c|c|} \hline x & n & p & Excel formula & Probability, P(x) {Round to three decimals} \\ \hline 3 & 8 & 0.6 & =BINOM.DIST(3, 8, 0.6, FALSE) & 0.124 \\ \hline \end{array}\)

Therefore, this probability is \(\displaystyle{P}{\left({3}\right)}={0.124}\)

Answer: 0.124

Step 2

b) Given: \(\displaystyle{n}={8},{p}={0.6},{x}={5}\) \(X\sim\)Binomial \(\displaystyle{\left({x}={8},{p}={0.6}\right)}\) I will use the excel formula "=BINOM.DIST(x, n, p, FALSE)" to find the probability.

\(\begin{array}{|c|c|} \hline x & n & p & Excel formula & Probability, P(x) {Round to three decimals} \\ \hline 5 & 8 & 0.6 & =BINOM.DIST(5, 8, 0.6, FALSE) & 0.279 \\ \hline \end{array}\)

Therefore, this probability is \(\displaystyle{P}{\left({5}\right)}={0.279}\)

Answer: 0.279

Step 3

c) Given: \(\displaystyle{n}={8},{p}={0.6}\)

\(X\sim\) Binomial \(\displaystyle{\left({n}={8},{p}={0.6}\right)}\) find: \(\displaystyle{P}{\left({3}\leq{X}\leq{5}\right)}={P}{\left({3}\right)}+{P}{\left({4}\right)}+{P}{\left({5}\right)}\) I will use the excel formula "=BINOM.DIST(x, n, p, FALSE)" to find the probability. \(\begin{array}{|c|c|} \hline x & n & p & Excel formula & Probability, P(x) Round\ to\ three\ decimals \\ \hline 3 & 8 & 0.6 & =BINOM.DIST(3, 8, 0.6, FALSE) & 0.124 \\ \hline 4 & 8 & 0.6 & =BINOM.DIST(4, 8, 0.6, FALSE) & 0.232 \\ \hline 5 & 8 & 0.6 & =BINOM.DIST(5, 8, 0.6, FALSE) & 0.279 \\ \hline \end{array}\)

Therefore, this probability is \(\displaystyle{P}{\left({3}\leq{X}\leq{5}\right)}={0.124}+{0.232}+{0.279}\)
\(\displaystyle{P}{\left({3}\leq{X}\leq{5}\right)}={0.635}\)

Answer: 0.635

Step 4

d) Given: \(\displaystyle{n}={12},{p}={0.1}\)

\(X\sim\) Binomial \(\displaystyle{\left({n}={12},{p}={0.1}\right)}\) find: \(\displaystyle{P}{\left({1}\leq{X}\right)}=?\) Use complement rule, \(\displaystyle{P}{\left({1}\leq{X}\right)}={1}-{P}{\left({0}\right)}\) I will use the excel formula "=BINOM.DIST(x, n, p, FALSE)" to find the probability. \(\begin{array}{|c|c|} \hline x & n & p & Excel formula & Probability, P(x) Round\ to\ three\ decimals \\ \hline 0 & 12 & 0.1 & =BINOM.DIST(0, 12, 0.1, FALSE) & 0.282 \\ \hline \end{array}\)

Therefore, this probability is \(\displaystyle{P}{\left({1}\leq{X}\right)}={1}-{0.282}\)
\(\displaystyle{P}{\left({1}\leq{X}\right)}={0.718}\) Answer: 0.718

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