# Probability Worksheets - Page 3

Probability Worksheets
• Page 3
21.
If the event B depends on the event A, then P(B|A) =
 a. b. c. d.

#### Solution:

If the event B depends on the event A, then P(B|A) = P(A and B)P(A).
[Conditional Probability Formula.]

22.
A card is drawn from a well-shuffled deck of 52 cards and then a second card is drawn. Find the probability that the first card is a spade and the second card is a club if the first card is not replaced.
 a. $\frac{13}{51}$ b. $\frac{13}{102}$ c. $\frac{51}{52}$ d. $\frac{13}{204}$

#### Solution:

P(first spade card) = P(S) = 13 / 52 = 1 / 4

After the event of drawing a spade, the deck has 51 cards, of which 13 are clubs(C).

Therefore, P(C|S) = 13 / 51

Hence, P(S and C) = P(S) · P(C|S)
[Conditional Probability.]

= (14) · (1351)

= 13204

23.
Two dice were thrown and it is known that the numbers which come up were different. Find the probability that the sum of the two numbers was 4.
 a. $\frac{1}{15}$ b. $\frac{4}{15}$ c. $\frac{1}{30}$ d. $\frac{2}{15}$

#### Solution:

Let A be the event in which the two dice show different numbers and let B be the event in which the sum is 4. Then P(A) = 30 / 36

There are two outcomes (3, 1) and (1, 3) in which the sum is 4 and the numbers are different. Hence, P(B A) = 2 / 36

P(B|A) = P(B ∩ A)P(A)
[Conditional probability.]

= ( 236) ÷ ( 3036)

= 115

24.
A dice is thrown 6 times. If 'getting an even number' is a success, then what is the probability of exactly 5 successes?
 a. $\frac{5}{32}$ b. $\frac{3}{32}$ c. $\frac{1}{16}$ d. $\frac{1}{10}$

#### Solution:

In a single trail, probability p of success is p = P(getting an even number 2, 4 or 6)

= 36 = 12

Probability of failure, q = 1 - p

= 1 - 12 = 12

P(r) = (nr) (p)r(q)n - r
[Binomial distribution.]

P(exactly 5 successes) = (65) (12)5(12)1

= 332

25.
A coin is tossed 5 times. If 'getting a head' is considered as a success, find the probability of at least 3 successes.
 a. $\frac{1}{6}$ b. $\frac{1}{2}$ c. $\frac{1}{4}$ d. $\frac{1}{8}$

#### Solution:

In a single trial, probability p of success is given by p = P(getting a head) = 1 / 2

Probability of failure, q = 1 - p = 1 - 1 / 2 = 1 / 2

P(r) = (nr) (p)r(q)n - r
[Binomial distribution.]

P(r) = p(3) + p(4) + p(5)

= (53)(1 / 2)3(1 / 2)2 + (54)(1 / 2)4(1 / 2) + (1 / 2)5

= 1032 + 532 + 132

= 1632 = 12

26.
A box contains 20 cards labeled 1 through 20. One card is drawn from it at random. What is the probability of getting an odd number and the odds in favor of getting an odd number?
 a. $\frac{2}{5}$ and 1 : 1 b. $\frac{1}{10}$ and 1 : 2 c. $\frac{9}{10}$ and 1 : 2 d. $\frac{1}{2}$ and 1 : 1

#### Solution:

P (getting an odd number) = 10 / 20 = 1 / 2
[There are 10 odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.]

Odds in favor of getting an odd number are 10 : 10 or 1 : 1
[Ten of the outcomes are successes and 10 of them are failures.]

27.
Fifteen cards labeled A through O are placed in a bag. The bag is shaken and one card is drawn at random. Find the probability of not drawing a vowel.
 a. $\frac{4}{15}$ b. $\frac{4}{5}$ c. $\frac{1}{2}$ d. $\frac{11}{15}$

#### Solution:

P (not getting a vowel) = 11 / 15
[There are 11 ways of not getting a vowel: B, C, D, F, G, H, J, K, L, M, N.]

28.
If a card is drawn from a bridge deck, what is the probability of getting a king?
 a. $\frac{1}{2}$ b. $\frac{1}{13}$ c. $\frac{1}{52}$ d. $\frac{1}{4}$

#### Solution:

Let E be the event of getting a king from the 4 in the deck.

Number of outcomes in E = 4C1

= 4! / 1!(4-3)!
[nCr = n! / r!(n-r)! .]

= 4×3! / 1!3! = 4

Let S be the sample space consists of all cards.

Number of outcomes in S = 52C1

= 52! / 1!(52-1)! = 52
[nCr = n! / r!(n-r)! .]

P(E) = Number of outcomes in the eventNumber of outcomes in the sample space

= 4 / 52= 1 / 13

29.
A box contains 20 cards labeled 1 through 20 on them. One card is drawn from it at random. Find the probability of getting an even number.
 a. $\frac{9}{20}$ b. $\frac{1}{2}$ c. $\frac{1}{20}$ d. $\frac{1}{4}$

#### Solution:

P (getting an even number) = 10 / 20 = 1 / 2
[There are 10 even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.]

30.
A box contains 20 cards labeled 1 through 20. One card is drawn from it at random. Find the probability of getting a prime number.
 a. $\frac{1}{2}$ b. $\frac{9}{20}$ c. $\frac{1}{5}$ d. $\frac{2}{5}$

#### Solution:

P (getting a prime number) = 8 / 20 = 2 / 5
[There are 8 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19.]