Question

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is

(i)    n black king
(ii)    either a black card or a king
(iii)    black and a king
(iv)    a jack, queen or a king
(v)    neither a heart nor a king
(vi)    spade or an ace

(vii)    neither an ace nor a king.

Solution

Total number of all possible outcomes = 52.
i.e., n( S) = 52
(i) Let A be the favourable outcomes of getting a black king, then
n( A) = 2
Therefore,
P(A) = $fraction numerator straight n left parenthesis straight A right parenthesis over denominator straight n left parenthesis straight S right parenthesis end fraction equals 2 over 52 equals 1 over 26$

(ii) Let B be the favourable outcomes of getting either a black card or a king, then n(B) = 28
Therefore,
P(B) = $fraction numerator straight n left parenthesis straight B right parenthesis over denominator straight n left parenthesis straight S right parenthesis end fraction equals 28 over 52 equals 7 over 13$

(iii) Let C be the favourable outcomes of getting black and a king, then n(C) = 2
Therefore,
P(C) = $fraction numerator straight n left parenthesis straight C right parenthesis over denominator straight n left parenthesis straight S right parenthesis end fraction equals 2 over 52 equals 1 over 26$

(iv) Let D be tine favourable outcomes of getting a jack, queen or a king then n(D) = 12
Therefore,
P(D) = $fraction numerator n left parenthesis D right parenthesis over denominator n left parenthesis S right parenthesis end fraction equals 12 over 52 equals 3 over 13$

(v) Let E be the favourable outcomes of getting neither a Heart nor a king, then n(E) = 36
Therefore,
P(E) = $fraction numerator straight n left parenthesis straight E right parenthesis over denominator straight n left parenthesis straight S right parenthesis end fraction equals 36 over 52 equals 9 over 13$

(vi) Let F be the favourable outcomes of getting spade or an ace, then
n(F) = 16
Therefore,
P(F) = $fraction numerator straight n left parenthesis straight F right parenthesis over denominator straight n left parenthesis straight S right parenthesis end fraction equals 16 over 52 equals 4 over 13$

(vii) Let G be the favourable outcomes of getting neither an ace nor a king, then n(G) = 44
Therefore,
P(G) = $fraction numerator straight n left parenthesis straight G right parenthesis over denominator straight n left parenthesis straight S right parenthesis end fraction equals 44 over 52 equals 11 over 13$

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