Video: Finding the Probability of Neither of Two Independent Events Occuring

𝐴 and 𝐡 are independent events, where 𝑃(𝐴) = 5/6 and 𝑃(𝐡) = 3/4. What is the probability that neither event 𝐴 nor event 𝐡 occurs?

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Video Transcript

𝐴 and 𝐡 are independent events where the probability of 𝐴 equals five-sixths and the probability of 𝐡 equals three-quarters. What is the probability that neither event 𝐴 nor event 𝐡 occurs.

In order to answer this question, we need to recall some of the facts about probability. The probability of an event not occurring, written 𝑃 of 𝐴 prime, is equal to one minus the probability of 𝐴 occurring. As the probability of 𝐴 is equal to five-sixths, the probability of 𝐴 not occurring is one minus five-sixths. This is equal to one-sixth, as one whole one is equal to six-sixths and six minus five equals one. We can repeat this process to calculate the probability of 𝐡 not occurring. This will be equal to one minus three-quarters. As there are four quarters in a whole one, one minus three-quarters is equal to one-quarter.

We’re also told in the question that 𝐴 and 𝐡 are independent events. This means that the probability of 𝐴 and 𝐡 or 𝐴 intersection 𝐡 is equal to the probability of 𝐴 multiplied by the probability of 𝐡. We can, therefore, calculate the probability that neither event 𝐴 nor event 𝐡 occurs by multiplying the probability of not 𝐴 by the probability of not 𝐡. We need to multiply one-sixth by one-quarter. When multiplying fractions, we multiply the numerators and separately the denominators. One multiplied by one is one and six multiplied by four is 24. We can, therefore, conclude that the probability that neither event 𝐴 nor event 𝐡 occurs is one out of 24 or one 24th.

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