A hand of three cards is chosen from a standard deck of 52 cards. You get Rs 3 for each heart in your hand. If X denotes the random variable measuring your gain, then find expected value of X.
Answers
Given : A hand of three cards is chosen from a standard deck of 52 cards. You get Rs 3 for each heart in your hand.
X denotes the random variable measuring your gain,
To find : expected value of X.
Step-by-step explanation:
X = 0 if no heart
X = 3 if 1 heart
X = 6 if 2 heart
X = 9 if 3 heart
P (0) = ¹³C₀*³⁹C₃/⁵²C₃ = 39 * 38 * 37 / 52 * 51 * 50 = 703/1700
P (3) = ¹³C₁*³⁹C₂/⁵²C₃ = 39 * 38 * 13 *3 / 52 * 51 * 50 =741/1700
P (6) = ¹³C₂*³⁹C₁/⁵²C₃ = 39 *13 * 12 *3 / 52 * 51 * 50 = 234/ 1700
P (0) = ¹³C₃*³⁹C₀/⁵²C₃ = 13 * 12 *11 / 52 * 51 * 50 = 22/1700
expected value of X = ∑xP(x)
= (0 * 703 + 3 * 741 +6 * 234 + 9 * 22)/1700
= (2223 + 1404 + 198)/1700
= 3,825 / 1700
= 2.25
expected value of X. = 2.25
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