a fair dice is rolled 3 times one after another Find the pro

a fair dice is rolled 3 times, one after another. Find the probability that at least one 5 is obtained but no 6\'s are rolled.

Solution

Number of ways of occurence of atleast one 5 and no 6:

When 5 occurs one time and 6 doesnt occur

= 3C1 (to choose which time 5 occurs out of 3 times) * (1/6) (for 5) * (4/6)² (for other than 5 and 6)

When 5 occurs two times and 6 doesnt occur

= 3C2 (to choose which times 5 occurs out of 3 times) * (1/6)² (for 5) * (4/6)(for other than 5 and 6)

When 5 occurs three times and 6 doesnt occur

= 3C3 (to choose which times 5 occurs out of 3 times) * (1/6)³ (for 5) * (4/6)⁰ (for other than 5 and 6)

= 61/216 = 0.2824