Question

2. Twenty-four percent of snake species are venomous. A zoologist selects 18 snake species at random and is interested in the number [of snake species] that are venomous.

   1. a) Define the random variable of interest, X.

   2. b) State the distribution of X. Show your work by checking the relevant criteria.

   3. c) What is the probability that exactly 4 are venomous?

   4. d) What is the probability that at least one is venomous?

   5. e) What is the probability that exactly 15 are not venomous?

   6. f) Determine the expected value.

Answer 1

a)Here X=number of snakes that are venomous..

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b)

X follows Binomial distribution with probability of success=p=0.24 and number of trials=n=18

where success indiacates the event of getting a snake species which is  venomous

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PMF of X is

$P(X=x)=\binom{18}{x}(0.24)^{x}(1-0.24)^{18-x};x=0,1,2,3,.....,18$

c)

the probability that exactly 4 are venomous=P(X=4)

$P(X=4)=\binom{18}{4}(0.24)^{4}(1-0.24)^{18-4}=0.217749(Answer)$

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d)

the probability that at least one is venomous=P(X>=1)

$P(X\geq 1)=1-P(X< 1)=1-P(X=0)=\binom{18}{0}(0.24)^{0}(1-0.24)^{18-0}=1-0.0072=0.9928(Answer)$

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e)

the probability that exactly 15 are not venomous=P(X=15)^c=1-P(X=15)

$P(X=15)^{c}=1-P(X=15)=1-\binom{18}{15}(0.24)^{15}(1-0.24)^{18-15}=1- 0.0000002=0.9999998(Answer)$

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f)

Expected value=E(X)=n*p=18*0.24=4.32

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