A box contains 50 index cards One letter is written on each

A box contains 50 index cards.

One letter is written on each card. There are
5 cards with the letter A
5 cards with the letter E
7 cards with the letter H
10 cards with the letter M
3 cards with the letter S
20 cards with the letter T

If one card is picked at random, determine

a) p(M,A,T, or H)

If one card is selected, replaced, and the second card selected, determine

a) p(bothA)

b) p(neither is a vowel)

If one card is selected, not replaced, and second card selected, determine

a) p(both A)

b) p(neither is a vowel)

Solution

If one card is picked at random

P(the card is M) = (10C1)/(50C1) = 10/50

P(the card is A) = (5C1)/(50C1) = 5/50

P(the card is T) = (20C1)/(50C1) = 20/50

P(the card is H) = (7C1)/(50C1) = 7/50

now, avobe 4 events are mutually exclusive.

so, P(the card is A, M, T or H) = (5+10+20+7)/50 = 21/25.

If one card is selected, replaced and the second card selected,

then P( 1st card is A) = 5/50

P( 2nd card is A) = 5/50

so, P( both cards are A) = (5*5)/(50*50) = 1/100 [since, both selection is independent]

P( neither is vowel) = P( 1st card is H, M, S ot and 2nd card is also H, M, S ,T)

P(1st card is H, M, S,or T) = (7+10+3+20)/50 = 4/5

P(2nd card is H, M, S,or T) = (7+10+3+20)/50 = 4/5

so, P(neither is vowel) = (4*4)/(5*5) = 16/25

If one card is selected, not replaced and second card selected, then at time of 2nd selection, the number of available cards is 49.

P(1st card is A) = 5/50

P(2nd card is A) = 4/49 [ after 1st card is A, remaining cards with A are 4, and remaining total cards are 49.]

so, P(both cards are A) = (5*4)/(50*49) = 2/245

out of 50 cards, 40 are non vowel.

P(1st card is not vowel) = 40/50 = 4/5

P(2nd card is not vowel) = 39/49 [ after selecting a non vowel card, remaining now vowel cards are 39, and total remaining cards are 49]

so, P( neither is vowel) = (4*39)/(5*49) = 156/256.