A bag contains 24 mangoes out of which six are bad. If 6 mangoes are selected randomly from the bag with replacement, find the probability that not more tha...

Answer Details

Because selection is with replacement, each draw is independent with a constant probability of a bad mango:

\[p=\frac{6}{24}=\frac14,\qquad q=1-p=\frac34\]

Let \(X\) be the number of bad mangoes in \(n=6\) draws; \(X\sim\text{Bin}(6,\tfrac14)\). We need \(P(X\le3)\), which is easiest via the complement \(P(X\le3)=1-P(X\ge4)\).

\[P(4)=\binom{6}{4}\left(\tfrac14\right)^{4}\left(\tfrac34\right)^{2}=\frac{15\times9}{4096}=\frac{135}{4096}\]

\[P(5)=\binom{6}{5}\left(\tfrac14\right)^{5}\left(\tfrac34\right)=\frac{6\times3}{4096}=\frac{18}{4096}\]

\[P(6)=\left(\tfrac14\right)^{6}=\frac{1}{4096}\]

\[P(X\ge4)=\frac{135+18+1}{4096}=\frac{154}{4096}=\frac{77}{2048}\]

\[P(X\le3)=1-\frac{77}{2048}=\frac{1971}{2048}\approx0.9624\]