Question

In Europe, 53% of flowers of the rewardless orchid, Dactylorhiza sambucina are yellow, whereas the remaining flowers are purple (Gigord et al. 2001). If we took a random sample of five individuals, what is the probability that at least three would be yellow? Include the leading zero, and state your answer to three decimal places.

Answer 1

Probability that at least three would be yellow = 0.556 .

Explanation:

We are given that In Europe, 53% of flowers of the reward less orchid, Dactylorhiza sambucina are yellow.

The Binomial distribution probability is given by;

P(X = r) =
`\binom{n}{r}p^(r)(1-p)^(n-r)` for x = 0,1,2,3,.......

Here, n = number of trials which is 5 in our case

r = no. of success which is at least 3 in our case

p = probability of success which is probability of yellow flower of

0.53 in our case

So, P(X >= 3) = P(X = 3) + P(X = 4) + P(X = 5)

=
`\binom{5}{3}0.53^(3)(1-0.53)^(5-3) + \binom{5}{4}0.53^(4)(1-0.53)^(5-4) + \binom{5}{5}0.53^(5)(1-0.53)^(5-5)`

=
`10*0.53^(3)(0.47)^(2) + 5*0.53^(4)(0.47)^(1) + 1*0.53^(5)(0.47)^(0)`

= 0.556

Therefore, the probability that at least three would be yellow is 0.556 .