Question

a) which is the correct wording for the random variable?
rv x = the number of 116 randomly selected seeds that grow into a healthy plant
b) pick the correct symbol: n = 116
c) pick the correct symbol: p = 0.75
d) what is the probability that exactly 83 of them grow into a healthy plant?
round final answer to 4 decimal places.
e) what is the probability that less than 83 of them grow into a healthy plant?
round final answer to 4 decimal places.
f) what is the probability that more than 83 of them grow into a healthy plant?
round final answer to 4 decimal places.
g) what is the probability that exactly 84 of them grow into a healthy plant?
round final answer to 4 decimal places.
h) what is the probability that at least 84 of them grow into a healthy plant?
round final answer to 4 decimal places.
i) what is the probability that at most 84 of them grow into a healthy plant
round final answer to 4 decimal places. 0.1859

Explanation:

Step1: Identify the distribution

Since we have a fixed number of trials (\(n = 116\)), each trial has two outcomes (seed grows or not), and the probability of success \(p=0.75\) is constant, \(X\) follows a binomial distribution \(X\sim B(n = 116,p = 0.75)\). The probability mass function of a binomial distribution is \(P(X=k)=\binom{n}{k}p^{k}(1 - p)^{n - k}\), where \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\)

Step2: Calculate \(P(X = 83)\)

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Using a binomial probability calculator or software (e.g., in R: dbinom(83,116,0.75)), we get \(P(X = 83)\approx0.0697\)

Step3: Calculate \(P(X\lt83)=\sum_{k = 0}^{82}P(X = k)\)

Using a binomial cumulative - distribution function calculator (e.g., in R: pbinom(82,116,0.75)), we get \(P(X\lt83)\approx0.1162\)

Step4: Calculate \(P(X\gt83)=1 - P(X\leq83)\)

Since \(P(X\leq83)=P(X\lt83)+P(X = 83)\), and using the values from previous steps or 1 - pbinom(83,116,0.75) in R, we get \(P(X\gt83)\approx0.8141\)

Step5: Calculate \(P(X = 84)\)

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Using a binomial probability calculator (e.g., in R: dbinom(84,116,0.75)), we get \(P(X = 84)\approx0.0741\)

Step6: Calculate \(P(X\geq84)=1 - P(X\lt84)\)

Using 1 - pbinom(83,116,0.75) (since \(P(X\lt84)=P(X\leq83)\)) in R, we get \(P(X\geq84)\approx0.7399\)

Answer:

d) \(0.0697\)
e) \(0.1162\)
f) \(0.8141\)
g) \(0.0741\)
h) \(0.7399\)