Question

Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use itto solve the problem. Use either a uniform or non-uniform sample space or try both.

A shopping mall has four entrances, one on the North, one on the South, and twoon the East. If you enter at random, shop and then exit at random, what is theprobability that you enter and exit on the same side of the mall?

Short answer

The required sample space is$\left(NN,NS,N{E}_1,N{E}_2,SN,SS,S{E}_1,S{E}_2,E_{1}N,E_{1}S,E_1{E}_1,E_1{E}_2,E_{2}N,E_{2}S,E_2{E}_1,E_2{E}_2\right)$

with the probability that one enters and exits on the same side of the mallis $\frac{3}{8}$.

Step-by-step solution

Significance of Probability

The probability is used to measure that a particular event will occur or not. The value of probability is always equal to less than 1. It can be obtained from the proper sample space.

Determination of the total number of possible outcomes.

There are two entrances on East side, so, both the entrances will be considered unique.

Let $N , S , E_1 , E_2$be the name of entrances in North, South and East directions respectively.

Any person can enter and exit through any doors and repetition is possible.

For entry and exits, there are 4 choices each and the sample space is expressed as follows,

localid="1655875190229">NN,NS,NE1,NE2,SN,SS,SE1,SE2,E1N,E1S,E1E1,E1E2,E2N,E2S,E2E1,E2E2

Determination of the probability that one enters and exits on the same side of the mall

Each point of the obtained sample space has an equal probability of$\frac{1}{16}$ .

The possible combination for entry and exits being on the same side are 6 namely .

$\left(NN , SS , E_1{E}_1 , E_1{E}_2 , E_2{E}_1 , E_2{E}_2\right)$

Find the probability that one enters and exits on the same side of the mall that is 6 times $\frac{1}{16}$.

$\begin{array}{rcl}p& =& 6\times \frac{1}{16}\\ & =& \frac{3}{8}\\ & & \end{array}$

Thus, the probability that one enters and exits on the same side of the mall is $\frac{3}{8}$.