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31. One card is drawn from a pack of 52 cards. What is the probability that the card drawn is either a red card or a king ?

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Solution:
Here, n(S) = 52
There are 26 red cards (including 2 kings) and there are 2 more kings.
Let E = event of getting a red card or a king.
Then, n(E) = 28
∴P(E)=n(E)n(S)=2852=713
32. Two cards are drawn from a pack of 52 cards. The probability that either both are red or both are king, is-

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Solution:
Clearly,
n (S) = n (S)=   52C2=   (52×51)2   = 1326
Let E1 = event of getting both red cards
E2 = event of getting both kings
Then, E1 ∩ E2 = event of getting 2 kings of red cards.
∴ n (E1)=26C2=(26×25)(2×1)     = 325 and
n (E2)=4C2=(4×3)(2×1)     = 6
n(E1∩E2)=2C2=1
∴P(E1)=n(E1)n(S)=3251326      and
P(E2)=n(E2)n(S)=61326
P(E1∩E2)=11326
∴ P (both red or both kings)
=P(E1∪E2)=P(E1)+P(E2)−P(E1∩E2)=(3251326+61326−11326)=3301326=55221
33. An urn contains 6 red, 4 blue, 2 green 3 yellow marbles. If two marbles are drawn at random from the run, what is the probability that both are red ?

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Solution:
Total number of balls = (6 + 4 + 2 + 3) = 15
Let E be the event of drawing 2 red balls.
Then, n(E)  =6C2  =6×52×1   = 15
Also, n(S)=15C2   =15×142×1   = 105
∴P(E)=n(E)n(S)=15105=17
34. A basket contains 4 red, 5 blue and 3 green marbles. If three marbles are picked up at random what is the probability that at least one is blue ?

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Solution:
Total number of marbles = (4 + 5 + 3) = 12
Let E be the event of drawing 3 marbles such that none is blue.
Then, n (E) = number of ways of drawing 3 marbles out of 7 = 7C3   =7×6×53×2×1   = 35
And, n(S)=12C3   =12×11×103×2×1   = 220
∴P(E)=n(E)n(S)=35220=744
∴ Required probability
= 1 - P(E)
= (1−744)
= 3744
35. A box contains 20 electric bulbs, out of which 4 are defective. Two balls are chosen at random from this box. The probability that at least one of them is defective, is -

Discuss
Solution:
P (none is defective)
= n (E) =  16C220C2= (16×152×1×2×120×19)     =1219
P (at least 1 is defective)
=(1−1219)   =719
36. An urn contains 6 red, 4 blue, 2 green and 3 yellow marbles. If two marbles are picked up at random, what is the probability that either both are green or both are yellow ?

Discuss
Solution:
Total number of marbles = (6 + 4 + 2 + 3) = 15
Let E be the event of drawing 2 marbles such that either both are green or both are yellow.
Then,
n (E) = (2C1+3C2)   =(1+3C1)   = (1 + 3) = 4
And,n (S) = 15C2=   15×142×1   = 105
∴P(E)=n(E)n(S)=4105
37. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5 ?

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Solution:
Here, S = {1, 2, 3, 4, ....., 19, 20}
Let E = event of getting a multiple of 3 or 5 = {3, 6, 9, 12, 15, 18, 5, 10, 20}
∴P(E)=n(E)n(S)=920
38. In a single throw of die, what is the probability of getting a number greater than 4 ?

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Solution:
When a die is thrown, we have S = {1, 2, 3, 4, 5, 6}
Let, E = event of getting a number greater than 4 = {5, 6}
∴P(E)=n(E)n(S)=26=13
39. One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card ?

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Solution:
Clearly, there are 52 cards, out of which there are 12 face cards 4 jack, 4 queens, and 4 kings
∴ P (getting a face card) =1252   =313
40. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even ?

Discuss
Solution:
In a simultaneous throw of two dice, we have n (S) = (6 × 6) = 36
Let E = event of getting two numbers whose product is even.
Then, E = {(1, 2), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 4), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 4), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
∴ n (E) = 27
∴P(E)=n(E)n(S)=2736=34

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