If X is a radom variable for amount you win, Then the excepted value of game or expected income i.e E(X) is 8/13.
Expected Income is the result of multiplying your income by the probability of earning it. This expected revenue can be subtracted by the expected cost to get the expected profit.
Mathematically, E(X) = X₁p₁ + X₂p₂ + ---+ Xₙpₙ
where Xᵢ are observation and pᵢ probability of Xᵢ.
We have given that,
Total number of cards in standard card deck = 52
i.e total possible outcomes = 52
One card is drawn from deck.
Total number of king cards in 52-card deck = 4
Favourable outcomes that drawn card is king = 4
Probability of getting a king card , p = 4/52 = 1/13
Probability of not getting a king ,q = 1 - p
= 1 - 1/13 = 12/13
let X be the amount you win . You win $20 if a king is drawn and lose $1 if king is not drawn . So, probability distribution is
Xᵢ pᵢ
$20 1/13
-$1 12/13
So, E(X) = ∑ Xᵢpᵢ = 20× 1/13 + (-1)× 12/13 = 20/13 - 1/13 = 8/13
Hence, required excepted value is 8/13.
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