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Question: 6 dice are thrown 729 times. How many times do you expect atleast 3 dice to show 4 or 5....

6 dice are thrown 729 times. How many times do you expect atleast 3 dice to show 4 or 5.

A

224

B

228

C

233

D

238

Answer

233

Explanation

Solution

n = 6, p = 13,q=23\frac { 1 } { 3 } , q = \frac { 2 } { 3 } .

P(x ≥ 3) = p (x = 3) + p (x=4) + p(x=5) + p(x=6)

= 6C3(13)3(23)3+6C4(13)4(23)2+6C5(13)5(23)+6C6(13)6{ } ^ { 6 } \mathrm { C } _ { 3 } \left( \frac { 1 } { 3 } \right) ^ { 3 } \left( \frac { 2 } { 3 } \right) ^ { 3 } + { } ^ { 6 } \mathrm { C } _ { 4 } \left( \frac { 1 } { 3 } \right) ^ { 4 } \left( \frac { 2 } { 3 } \right) ^ { 2 } + { } ^ { 6 } \mathrm { C } _ { 5 } \left( \frac { 1 } { 3 } \right) ^ { 5 } \left( \frac { 2 } { 3 } \right) + { } ^ { 6 } \mathrm { C } _ { 6 } \left( \frac { 1 } { 3 } \right) ^ { 6 }

= 20×836+15×436+1236+136=23336\frac { 20 \times 8 } { 3 ^ { 6 } } + \frac { 15 \times 4 } { 3 ^ { 6 } } + \frac { 12 } { 3 ^ { 6 } } + \frac { 1 } { 3 ^ { 6 } } = \frac { 233 } { 3 ^ { 6 } }

= 233729\frac { 233 } { 729 }

Frequency of success = 133729×729\frac { 133 } { 729 } \times 729 =233