Equally Likely Outcomes Examples

# Equally Likely Outcomes Examples

The events having same probability of their occurrence are known as equally likely outcomes. Tossing of coin, rolling of dice give outcomes which are all having equal probability and they are equally likely outcomes. The probability of any event is the ratio of number of favorable events to total number of events. If a dice is rolled, and all six faces have six different numbers on them then the probability of getting any number equals the probability of getting the other numbers. Hence, all six numbers will be called equally likely outcomes. If for any two events, their probabilities are different then they cannot be called equally likely events.

## Word Problems

Let us have a look on few examples for better understanding of the concept.
Problem 1:

Two cards are being chosen from a deck of $52$ cards. Find the probability that both are Jack.

Solution:

The probability of first card being jack = $\frac{4}{52}$

Now, there are $51$ cards where $3$ kings are present.

The probability of second card being jack = $\frac{3}{51}$

The probability that both of them are jack = $\frac{4}{52}$ $\times$ $\frac{3}{51}$ = $\frac{1}{221}$
Problem 2:

A dice has six sides out of which two sides have $6$ written on them, and other four sides have $1,\ 2,\ 3,\ 4$ written on them. Which all are equally likely outcomes when the dice is rolled?

Solution:

The set of outcomes = $\{1,\ 2,\ 3,\ 4,\ 6\}$

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

Probability of getting a $2$ = $\frac{1}{6}$

Probability of getting a $3$ = $\frac{1}{6}$

Probability of getting a $4$ = $\frac{1}{6}$

Probability of getting a $6$ = $\frac{2}{6}$ = $\frac{1}{3}$

The outcomes with same probability are $1,\ 2,\ 3,\ 4$.

Hence, $\{1,\ 2,\ 3,\ 4\}$ are equally likely outcomes.
Problem 3:

There is bag having $2$ red, $3$ green and $3$ yellow balls. Out of the given events which are equally likely?

a) Getting a green ball and getting a yellow ball.

b) Getting a red ball and getting a yellow ball.

Solution:

a) The probability of getting a green ball = $\frac{3}{8}$

The probability of getting a yellow ball = $\frac{3}{8}$

As the probabilities of both the events are same, they are equally likely.

b) The probability of getting a red ball = $\frac{2}{8}$

The probability of getting a yellow ball = $\frac{3}{8}$

As the probabilities of both the events are not same, they are not equally likely.
Problem 4:

A game generates a random number from $1$ to $100$. Which of the given events are equally likely?

a) Getting an even number

b) Getting an odd number

c) Getting a prime number

d) Getting a three digit number

Solution:

The probabilities of the events are:

a) Getting an even number = $\frac{50}{100}$ = $\frac{1}{2}$

b) Getting an odd number = $\frac{50}{100}$ = $\frac{1}{2}$

c) Getting an even number = $\frac{25}{100}$ = $\frac{1}{4}$

d) Getting an even number = $\frac{1}{100}$

Only getting an odd number and getting even number has equal probabilities. Hence, these events are equally likely.