5 bad apples are mixed with 10 good ones. If 3 apples are drawn one by

5 bad apples are mixed with 10 good ones. If 3 apples are drawn one by one with replacement, then find the probability distribution of the number of good apples.

Answer»

There are 5 bad apples and 10 good apples.

\(\therefore\) Total number of apples = 5 + 10 = 15

Let X denotes the number of good apples in 3 drawn apples.

Probability of success = \(\frac{Numbere\,of\,good\,apples}{Total number\,of\,apples}\) = 10/15 = 2/3

\(\therefore\) P = 2/3

Probability of failure is q = 1 - p = 1 - 2/3 = 1/3

P(x = 0) = ³C₀p⁰q³ = 1 x 1 x \((1/3)^3\) = 1/27

P(x = 1) = ³C₁p¹q² = 3 x 2/3 x (1/3)² = 2/9

P(x = 2) = ³C₂p²q = 3 x (2/3)² x 1/3 = 4/9

P(x = 3) = ³C₃p³q⁰ = 1 x (2/3)³ x 1 = 8/27

Hence, probability distribution of the number of good apple

X	0	1	2	3
P(X)	1/27	2/9	4/9	8/27

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