Problem:
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. The probability that X ≤ 3 equals [JEE 2009]

Solution:
To find the probability that X ≤ 3, we need to calculate the probability of getting a six in the first, second, or third toss.

Step 1: Probability of getting a six in the first toss
The probability of getting a six in any single toss of a fair die is 1/6. Therefore, the probability of getting a six in the first toss is 1/6.

Step 2: Probability of not getting a six in the first toss
The probability of not getting a six in any single toss of a fair die is 5/6. Therefore, the probability of not getting a six in the first toss is 5/6.

Step 3: Probability of getting a six in the second toss
If we did not get a six in the first toss, then we need to calculate the probability of getting a six in the second toss. The probability of getting a six in any single toss of a fair die is still 1/6. Therefore, the probability of getting a six in the second toss, given that we did not get it in the first toss, is also 1/6.

Step 4: Probability of not getting a six in the first two tosses
Since the events of not getting a six in the first toss and not getting a six in the second toss are independent, we can multiply their probabilities to calculate the probability of not getting a six in the first two tosses. Therefore, the probability of not getting a six in the first two tosses is (5/6) * (5/6) = 25/36.

Step 5: Probability of getting a six in the third toss
If we did not get a six in the first two tosses, then we need to calculate the probability of getting a six in the third toss. The probability of getting a six in any single toss of a fair die is still 1/6. Therefore, the probability of getting a six in the third toss, given that we did not get it in the first two tosses, is also 1/6.

Step 6: Probability of not getting a six in the first three tosses
Since the events of not getting a six in the first two tosses and not getting a six in the third toss are independent, we can multiply their probabilities to calculate the probability of not getting a six in the first three tosses. Therefore, the probability of not getting a six in the first three tosses is (25/36) * (5/6) = 125/216.

Step 7: Probability of X ≤ 3
The probability of X ≤ 3 is the probability of getting a six in the first toss (1/6), or not getting a six in the first toss and getting a six in the second toss (25/36 * 1/6), or not getting a six in the first two tosses and getting a six in the third toss (125/216 * 1/6). Therefore, the probability of X ≤ 3 is