Question Video: Finding the Probability of Two Events Occuring Together Using the Conditional Probability of One given the Other | Nagwa Question Video: Finding the Probability of Two Events Occuring Together Using the Conditional Probability of One given the Other | Nagwa

# Question Video: Finding the Probability of Two Events Occuring Together Using the Conditional Probability of One given the Other Mathematics • Third Year of Secondary School

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The probability that event π΄ occurs is 3/5. If event π΄ does not occur, then the probability of event π΅ occurring is 2/3. What is the probability that event π΄ does not occur and event π΅ occurs?

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### Video Transcript

The probability that event π΄ occurs is three-fifths. If event π΄ does not occur, then the probability of event π΅ occurring is two-thirds. What is the probability that event π΄ does not occur and event π΅ occurs?

Weβre given two pieces of information in this question. The first is that the probability of event π΄ occurring is three-fifths. This is straightforward, and we can write this π of π΄ is equal to three over five. The second piece of information is a conditional statement. This says that if event π΄ does not occur, then the probability of event π΅ occurring is two-thirds. And we write this as the probability of π΅ given not π΄ is equal to two-thirds. Remember, not π΄ is the complement of π΄ and this is written as π΄ with a bar on the top. And remember also that since the sum of all probabilities must equal one and that either π΄ or not π΄ must occur, then the probability of not π΄ is one minus the probability of π΄. And this will come in useful later on.

So we have the probability of π΄ is three-fifths and the probability of π΅ given not π΄ is two-thirds. The vertical bar in this expression reads given not, which is the conditional. Weβre asked to find the probability of not π΄ and π΅. Thatβs the probability that event π΄ does not occur and event π΅ does occur. And this symbol that looks like an n represents βandβ or the intersection of two events. So we have a simple probability, a conditional probability, and weβd like to find an intersection. And we can use the formula for conditional probability to find our intersection. This formula tells us that for two events πΆ and π·, the probability of πΆ given that π· has occurred is the probability of πΆ and π· divided by the probability of π·.

Now, if we let πΆ in the formula correspond to our event π΅ and π· in the formula is our event not π΄, then we can rewrite the formula as the probability of π΅ given not π΄ is equal to the probability of π΅ and not π΄ over the probability of not π΄. In our numerator, we know that the probability of π΅ intersection not π΄ is the same as the probability of not π΄ intersection π΅. So we can switch our π΅ and our not π΄ in the probability in the numerator.

And we see now that the numerator is actually what weβre looking for. Thatβs the probability of not π΄ intersection π΅. And we can make this the subject of our equation by multiplying through by the probability of not π΄. We can cancel this through on our right-hand side because the probability of not π΄ divided by the probability of not π΄ is equal to one so that we have the probability of not π΄ times the probability of π΅ given not π΄ is equal to the probability of not π΄ intersection π΅.

Now we know that the probability of π΅ given not π΄ is two-thirds. We also know that the probability of π΄ is three-fifths. And remembering that the probability of not π΄ is one minus the probability of π΄, we have the probability of not π΄ is one minus three-fifths, which is two-fifths. So in our formula, we have that the probability of not π΄ intersection π΅ is two-fifths times two-thirds. Multiplying fractions, we simply multiply the numerators and the denominators so that our numeratorβs two times two is equal to four and our denominator is five times three, which is 15, so that the probability that event π΄ does not occur and event π΅ occurs is four out of 15.

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