Molly is playing a game

[pamela]

WT book, chapter 5 probabilities, page 87.
Example: Molly is playing a game that requires her to roll a die repeatedly until she rolls a 1, at which point she must stop rolling the die. What's probability that she will roll the die less than four times before stopping?

My question is regarding the explanation, 1/6 + (5/6*1/6) + (5/6*5/6*1/6). I don't understand why the "5/6" needs to be used in our answer. Probability on first roll is 1/6. If a 1 is rolled here, we don't get to the next roll, so I would think it's 1/6*1/6*1/6 (if it's not rolled the first time, probability is 1/6 on the second one, and again 1/6 on the first roll).

Could you explain please?

[jnelson0612]

Sure! Okay, we want to know what the probability is that Molly can stop rolling less than four times; in other words, what is the probability that she rolls a 1 on either the first roll, the second roll, or the third roll? Since any of these possibilities are acceptable we will add them all together.

Chance of rolling a 1 on the first roll: 1/6

Chance of NOT rolling a 1 on the first roll, but then rolling a 1 on the second roll:
5/6 (chance of not rolling a 1 on the first roll) * 1/6 (chance of rolling a 1 on the second roll) = 5/6 * 1/6

Chance of NOT rolling a 1 on the first or second roll, but then rolling a 1 on the third roll:
5/6 (chance of not rolling a 1 on the first roll) * 5/6 (chance of not rolling a 1 on the second roll) * 1/6 (chance of rolling a 1 on the third roll) = 5/6 * 5/6 * 1/6

Thus, when I sum these probabilities together, I get 1/6 + (5/6*1/6) + (5/6*5/6*1/6).

Hope this makes more sense!