In a group of 96 people, 34 out of the 71 women have a smartphone, and 18 men do not have a smartphone. Determine the probability that a randomly picked smartphone owner in this group will be female.
We’re given a lot of information in this question about the numbers of men and women and whether or not they own a smartphone. Let’s organize this into a two-way table. So we have rows for men and women and columns for smartphone or no smartphone.
First of all, we’re told that the group consists of 96 people, so that is our overall total. We’re then told that there are 71 women, so this is our total for the row of women. Over these 71 women, we’re told that 34 of them have a smartphone. We can then work out the number of women that don’t have a smartphone by subtracting 34 from 71. So we have a complete row for women in the table.
Now, let’s look at the men. We’re told that 18 men do not have a smartphone. So we can fill this in in the table. We haven’t been given any other information in the question, but we can use the totals in the table to work out all of the other cells. If the group consists of 96 people and 71 of these are women, then we can work out the total number of men by subtracting 71 from 96. There are 25 men. If there are 25 men and 18 of them don’t own a smartphone, then the number that do can be found by subtracting 18 from 25. So we have seven men who do own a smartphone.
Finally, the totals can be found by summing the values in the columns. So we have 41 people whether men or women who do own smartphones and 55 who don’t. A sensible check here is to make sure that our column totals do add up to the group total, and they do; 41 plus 55 is 96.
So now, let’s consider how we can use this table to answer the question that we’ve been asked: determine the probability that a randomly picked smartphone owner in this group will be female. This is an example of a conditional probability. We know that the person chosen is a smartphone owner, and we want to work out the probability that they’re also female.
Let’s recall the conditional probability formula. The probability of 𝐵 given 𝐴 is the probability of the intersection of 𝐴 and 𝐵; that’s 𝐴 and 𝐵 divided by the probability of 𝐴. This tells us how to work out the conditional probability of an event 𝐵 happening given that event 𝐴 has already occurred.
In our question, the event that we know has already happened is the fact that this person is a smartphone owner and the event we are looking to find the conditional probability of is that they’re female. So substituting the terms SP, smartphone, and female for 𝐴 and 𝐵 in the formula tells us that the probability this person is female given that the owner has smartphone can be calculated by finding the probability they own a smartphone and a female and dividing it by the probability they own a smartphone.
Let’s look at each of these probabilities in the table. There are 34 women who own a smartphone. Therefore, the probability that this person owns a smartphone and is female will be 34 out of the total 96. The total of the people who own a smartphone whether they are men or women is 41. Therefore, the probability of somebody owning a smartphone is 41 out of the total 96.
So we have that the probability that this person is female given that they own smartphone is 34 over 96 divided by 41 over 96. The divisions by 96 in both the numerator and denominator of this overall fraction will cancel each other out. Therefore, we have our answer to the problem. The probability that a randomly picked smartphone owner in this group will be female is 34 over 41.