Multiple choice

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :

  1. $\dfrac { 150 } { 6 ^ { 5 } }$
  2. $\dfrac { 175 } { 6 ^ { 5 } }$
  3. $\dfrac { 200 } { 6 ^ { 5 } }$
  4. $\dfrac { 225 } { 6 ^ { 5 } }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To end on the 5th throw, the 5th throw must be a 4, and the 4th throw must be a 4. The first 3 throws must not contain the sequence '44'. Total outcomes = 6^5. Favorable outcomes: (not 4, not 4, not 4, 4, 4) + (4, not 4, not 4, 4, 4) + (not 4, 4, not 4, 4, 4). Calculating these sequences: 5*5*5 + 1*5*5 + 5*1*5 = 125 + 25 + 25 = 175.

AI explanation

The experiment ends on the fifth throw if the fourth and fifth rolls are both fours (probability 1/36), and no consecutive fours occur in the first three rolls. The valid combinations for the first three rolls that lack consecutive fours are 125 in number. Multiplying these valid sequences by the probability of the final two fours gives 125/36, which yields a total probability of 175 / 6 ^ 5.

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