# Video: Solving Problems Involving Probability

A box contains 56 balls. The probability of selecting at random a red ball is 5/7. How many balls in the box are not red?

02:24

### Video Transcript

A box contains 56 balls. The probability of selecting at random a red ball is five-sevenths. How many balls in the box are not red?

The event of selecting a red ball and the event of selecting a ball that is not red are complementary events. The probability of the complement of 𝐴, denoted 𝐴 bar, is equal to one minus the probability of 𝐴. In this question, we are told that the probability of selecting a red ball, 𝑃 of 𝑅, is equal to five-sevenths. This means that the probability of selecting a ball that is not red is one minus five-sevenths. This is equal to two-sevenths. The probability of an event occurring and its complement must always sum to one.

We are also told in this question that there are 56 balls in the box. Two-sevenths of these 56 balls are not red, so we need to calculate two-sevenths of 56. As the word “of” in mathematics means multiply, we need to multiply two-sevenths by 56. 56 is the same as 56 over one. We can then cross cancel or cross simplify by dividing 56 and seven by seven. This gives us two over one multiplied by eight over one, which is equal to 16 over one. We multiply the numerators and denominators separately. As this is equal to 16, we can conclude that 16 of the 56 balls in the bag are not red.

An alternative method here would be to calculate the number of red balls first. We can do this by working out five-sevenths of 56. This is equal to 40, so we have 40 red balls in the box. This means that the rest of the balls must not be red. 56 minus 40 is equal to 16. This once again proves that there are 16 balls in the box that are not red.