Probability of 2nd or 3rd Success on a Particular Attempt | Binomial Distribution | A-level Math S1

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[00:00:00]This is the question type that is getting repeated a lot in recent S1 papers where they ask you to find the probability of obtaining the second or third success on a particular attempt.

[00:00:09]Let's understand how to do this. In this example, we're given eight letters written on the sides of an eight-sided dice. We want to find the probability of obtaining the second A on the sixth roll of the dice. Now, how can you do this?

[00:00:21]Now, first of all, there are two A's, so the probability of getting an A on one attempt, that is two out of eight or one over four. Since we want the second success to come on the sixth attempt, what that really means is we want exactly one success in the first five attempts, and then the second success should be on the sixth attempt. Now, the first five therefore we can treat as binomial distribution. We could say n is equal to five. We want to find the probability of get of getting one success out of five attempts, and that will be this probability, and then multiply that with the probability of obtaining another success on the sixth attempt, and that will give you the final result.

[00:00:57]Now, here's another example. Let's suppose we have a fair six-sided dice, and we want to find the probability that a six is obtained for the third time on the seventh throw. It will be very similar to the previous example.

[00:01:07]If we want the third success to come on the seventh attempt, what that means is we want exactly two successes in the first six attempts. So, we treat the first six attempts as a binomial distribution, and we say find the probability of obtaining two successes out of six attempts, and then multiply that with another success probability for the seventh attempt, and that will give you the result. Now, sometimes they make it slightly more complicated.

[00:01:30]In this example for instance, they want us to find the probability that a three is obtained for the second time before the sixth throw. Now, before the sixth throw means the second success could come on the fifth attempt, on the fourth attempt, on the third attempt, or on the second attempt. So, we would have to consider all of those cases, add them up together, and that will give us the result. Now, another slightly more efficient way of doing this could be that we say, since we want to find the probability that the three is obtained for the second time before the sixth row, we could find the reverse probability. We find the probability that the second success does not come before the sixth row. So, what that means is we want the probability that we have less than two threes until the fifth row. So, it could be no three or one three. Find those two probabilities, probability of of obtaining only one three in the first five attempts or getting no three in the first five attempts, and then subtract that from one, and that will give you the probability that they're looking for, and that will be the final answer.

[00:02:35]I hope that helps. You'll find a lot of other revision content on my YouTube channel as well.

[00:02:39]I'll see you in the next video, inshallah.