Question

Suppose you randomly select and buy 4 tangerines from the grocery store, which had 12 tangerines on the shelf. Of the 12 tangerines, 8 had blemishes. What is the probability that two of your four tangerines will have blemishes?

Answer 1

Solution

Let X = number of tangerines out of 4 selected tangerines that have blemishes. Then,

X ~ B(4, p), where p = probability of one tangerine with blemishes, which is also equal to the proportion of tangerines with blemishes in the grocery store. So, p = 8/12 = 2/3 and thus,

X ~ B(4, 2/3) ………………………………………………………………………................................................……………… (1)

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then   

probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n …………………….…...........................................……………..(2)

[This probability can also be directly obtained using Excel Function: Statistical, INOMDIST……………………………….(2a)

Now to work out the solution,

Probability that two of the four tangerines selected will have blemishes

P(X = 2)

= (⁴C₂)(2/3)²(1/3)² [vide (1) and (2)]

= 8/27

= 0.2963 Answer

DONE