Compound events are combination of two or more simple events. Rolling a dice is a simple event but rolling a dice and choosing a card simultaneously is a compound event.To find the sample space and number of favorable events, organized lists, tree diagrams or tables can be used.

$P(E)$ = $\frac{n(E)}{n(S)}$

Example 1:

If a dice is rolled and a coin is tossed then what is the probability of getting a head and a 6?

Solution:

Sample space, $S = \{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)\}$

$n(S) = 12$

$E = \{(H, 1)\}$

$n(E) = 1$

$P(E)$ = $\frac{1}{12}$

Compound Probability Word Problems

Problem 1:

If a coin is tossed and a dice is rolled, then find the probability of getting an odd number and a tail using a tree diagram.

Solution:

First we see there are two possibilities, getting a head and a tail. Then, the tree diagram shows the possibility of getting numbers 1 to 6 in each case.

Compound Probability Word Problem

Number of leaves = Total number of possible outcomes = 12

Events having an odd number and a tail = 3

Probability of getting an odd number and a tail = $\frac{3}{12}$ = $\frac{1}{4}$
Problem 2:

Raashi has two books on mathematics, A and B, and four notebooks 3, 4, 5 and 6. If she takes out one book and one notebook, find the probability that it is book B and notebook 5 using a table.

Solution:

We can see the sample space using a table.

   3  4  5  6
A
A-3
A-4
A-5
A-6
B-3
B-4
B-5
B-6

Total number of possible events = 8

Events getting a B and a 5 together = 1

Probability of getting a B and a 5 together = $\frac{1}{5}$
Problem 3:

If a dice is rolled then what is the probability of getting a 4 or a 6?

Solution:


As they are exclusive compound events the probability will be sum of their individual probabilities.

$P(4)$ = $\frac{1}{6}$

$P(6)$ = $\frac{2}{6}$

Compound probability = $P(4)$ + $P(6)$ = $\frac{3}{6}$ = $\frac{1}{2}$
Problem 4:

If three coins are tossed simultaneously then what will be the probability of getting exactly two heads?

Solution:


Using a tree diagram we can see the sample of tossing three coins.

Compound Probability Word Problems

As the leaves are 8, the number of possible outcomes is 8.

Favorable outcomes = 3

Probability of getting two heads = $\frac{3}{8}$
Problem 5:

Jenny goes to a shop and likes three soft toys. One is teddy bear, other is tweety bird and third one is a Donald duck. She has to choose two toys, and the first one she has chosen as tweety bird. Find the probability of choosing Donald duck as second toy.

Solution:


First toy is tweety bird. Second toy has to options to be chosen from.

Probability of choosing a Donald Duck = $\frac{1}{2}$