If 6 cards are drawn from a deck of cards with replacement. What is the probability that the outcome is a club at least once. I thought it would be 13/52. But I was wrong.
$\endgroup$ 21 Answer
$\begingroup$If you don't draw a club at least once, you draw zero.
The odds of not drawing a club on a particular draw is $\frac{3}{4}$. Since you replace the card each time, each of the draws is from a full deck, and so the events are independent. Thus, the chance of drawing a nonclub 6 times is
$$P(\text{no clubs}) = \underbrace{\left(\frac{3}{4}\right) \cdots \left(\frac{3}{4}\right)}_n = \left(\frac{3}{4}\right)^6 = \frac{729}{4096},$$
and so the chance of drawing a least one club in $6$ draws is
$$P(\text{at least $1$ club}) = 1 - P(\text{no clubs}) = 1 - \frac{729}{4096} = \frac{3367}{4096} = 0.82202\ldots .$$
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