Question Video: Finding the Probability of a Difference of Two Events

A ball is drawn at random from a bag containing 12 balls each with a unique number from 1 to 12. Suppose 𝐴 is the event of drawing an odd number and 𝐵 is the event of drawing a prime number. Find P(𝐴 − 𝐵).

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Video Transcript

A ball is drawn at random from a bag containing 12 balls, each with a unique number from one to 12. Suppose 𝐴 is the event of drawing an odd number and 𝐵 is the event of drawing a prime number. Find the probability of 𝐴 minus 𝐵.

We are told in the question that there are 12 balls numbered from one to 12 in the bag. Each of these is equally likely to be drawn at random having a probability of one over 12 or one twelfth of being picked. We are told that 𝐴 is the event of drawing an odd number. There are six of these balls, numbers one, three, five, seven, nine, and 11. This means that the probability of event 𝐴 occurring is six over 12. By dividing the numerator and denominator by six, we see that this simplifies to one-half. We are also told that 𝐵 is the event of drawing a prime number. We know that a prime number has exactly two factors: the number one and the number itself. The prime numbers between one and 12 inclusive are two, three, five, seven, and 11. This means that the probability of event 𝐵 occurring is five over 12 or five twelfths.

We can also see that the probability of event 𝐴 and 𝐵 occurring, the probability of 𝐴 intersection 𝐵, is equal to four over 12. This is because four of the numbers, three, five, seven, and 11, are both odd and prime. We are asked to find the probability of 𝐴 minus 𝐵. Recalling the difference rule for probability, we know that the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. The probability of 𝐴 minus 𝐵 is therefore equal to a half or six twelfths minus four twelfths. This is equal to two twelfths which in turn simplifies to one over six or one-sixth. The probability of 𝐴 minus 𝐵 is equal to one-sixth.

We could actually have worked out this answer directly from our diagram. The probability of 𝐴 minus 𝐵 means we need to find odd numbers that are not prime. In this question, these are the numbers one and nine. Two out of the 12 numbers are odd and not prime. This confirms our answer of two over 12 or one-sixth.

Before considering our last two examples, we need to recall some other key probability formulae. The addition rule of probability states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. This formula can also be rearranged as shown. The complement of an event denoted 𝐴 bar or 𝐴 prime is the set of outcomes that are not the event. As probabilities sum to one, we know that the probability of the complement of 𝐴 is equal to one minus the probability of 𝐴. We will now use these two formulae together with the difference rule for probability to solve our last two examples.

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