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Six cards are drawn at random from a pack of 52 cards. Find the probability that at least 3 are hearts. |
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Answer» p = probability of getting a heart = \(\frac{13}{52} = \frac14\) q = 1 - p \(= 1 - \frac14 = \frac34\) n = 6 ∵ P(x = r) = \(^n C_r p^rq^{n -r}\) ∴ P(x = r) = \(^6 C_r p^rq^{6 -r}\) Required probability = \(P(x\ge 3)\) = P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) \(=\, ^6 C_3 \left(\frac14\right)^3 \left(\frac34\right)^3 +\, ^6C_4\left(\frac14\right)^4 \left(\frac34\right)^2 + \,^6C_5\left(\frac14\right)^5\left(\frac34\right)+\,^6C_6\left(\frac14\right)^6\) \(= \frac{6.5.4}{3.2.1}\times \frac{27}{4^6}+ \frac{6.5}{2.1}\times \frac9{4^6}+ 6\times \frac{3}{4^6}+ \frac1{4^6}\) \(= \frac1{4^6}(540 + 135 + 18 + 1)\) \(= \frac{694}{4096} \) \(= \frac{347}{2048}\) |
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