An event which remains unaffected by previous event or set of events is known as an independent event. For example, tossing a coin is an independent event as each time the coin is being tossed the result will be independent of the previous toss. If A and B are independent events then the probability of both the events happening will be $P(A\ and\ B) = P(A)*P(B)$, that is, the product of probability of each event.
Example 1:

A coin is being tossed twice. Find the probability of getting both heads.

Solution:

Probability of getting a head in one toss = 0.5

Probability of getting a head in second toss = 0.5

Probability of getting both heads = $0.5\times 0.5 = 0.25$.
Example 2:

A dice is being thrown twice. Find the probability of getting 5 and 6 in both throws respectively.

Solution:

Probability of getting a 5 in first throw = $\frac{1}{6}$

Probability of getting a 6 in second throw = $\frac{1}{6}$

Probability of both the events happening = $\frac{1}{6}\times \frac{1}{6}$ = $\frac{1}{36}$.


Independent Events Word Problems

Problem 1:

On a library shelf, three geometry and five algebra books. Books are not replaced after someone borrow it. If two books are taken then what is the probability that first is of geometry and other of algebra.

Solution:

The probability of first book to be of geometry = $\frac{3}{8}$

The probability of second book to be of algebra = $\frac{5}{7}$

As in total number of books one is already taken out, the total has become 7.

Probability of both events occurring = $\frac{3}{8}\times \frac{5}{7}$ = $\frac{15}{56}$.
Problem 2:

An exam is being conducted in two slots on Saturday and Sunday. Find the probability that Rose gets evening slot on Sunday.

Solution:

Probability of getting an evening slot out of two slots = 0.5

Probability of getting a slot on Sunday out of two days = 0.5

Probability of getting an evening slot on Sunday = $0.5 \times 0.5$ = $0.25$
Problem 3:

A coin is tosses and a dice is rolled simultaneously. What is the probability of a tail and a 6 coming as the output?

Solution:


Probability of a tail = $\frac{1}{2}$

Probability of a 6 coming as output = $\frac{1}{6}$

Probability of a tail coming in the toss and getting a 6 in dice = $\frac{1}{2}\times \frac{1}{6}$ = $\frac{1}{12}$.
Problem 4:

From a deck of 52 cards, two cards are being chosen without replacement. What will be the probability of first one being black and second one being red?

Solution:

Probability of first card being black = $\frac{26}{52}$ = $\frac{1}{2}$

Probability of second card being red = $\frac{26}{51}$

Probability of first card being black and second being red = $\frac{1}{2}\times \frac{26}{51}$ = $\frac{26}{102}$ = $\frac{13}{51}$
Problem 5:

The probability of Aren to solve a problem is 0.3 and the probability of Alice to solve the same problem is 0.6. Find the probability that the problem is solved by none of them.

Solution:


Probability that Aren cannot solve the problem = 1 - 0.3 = 0.7

Probability that Alice cannot solve the problem = 1 - 0.6 = 0.4

Probability that the problem is not solved = $0.7\times 0.4$ = 0.28