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S2

The question is: A fair dice is rolled 8 times. Find the probability of

(b) only 3 sixes

In the picture, I understand the fact that

P(3 sixes) = P(3S and 5S' in any order)
=

(\frac{1}{6})^3(\frac{5}{6})^5

as that would be the probabilities multiplied to their self the amount of times that they occur. However, they then used the formula for n choose k, which by my understanding only gives the total number of ways it can be arranged. So why was it multiplied to the probabilities in this example?

(edited 7 years ago)

Reply 1

NotNotBatman

Because getting a 6 and not getting a 6 can occur in any order.

If 8 dice were thrown simultaneously then the binomial coefficient wouldn't be included.

Reply 2

Naruke

Original post by NotNotBatman

Because getting a 6 and not getting a 6 can occur in any order.

If 8 dice were thrown simultaneously then the binomial coefficient wouldn't be included.

But why does order matter when they only ask for the probability????????????????

Reply 3

NotNotBatman

Original post by Naruke

But why does order matter when they only ask for the probability????????????????

If you had a tree diagram,for the probability you would total the number of paths through the three diagram. That's what the binomial coefficient works out.

Examsolutions explains it well. http://www.examsolutions.net/maths-revision/statistics/discrete-random-variables/binomial-distribution/formula/formula.php

@Zacken

help?

Original post by Naruke

@Zacken

help?

Label the throws from 1 to 6. You've found

P(1st throw is a 6) * P(2nd throw is a 6) * P(3rd throw is a 6) * P(4th throw isn't a 6) * P(5th throw isn't a 6) * P(6th throw isn't a 6) * ...

But you've ignored the possibilities that

P(1st throw isn't a 6) * P(2nd throw is a 6) * P(3rd throw is a 6) * P(4th throw is a 6) * P(5th throw isn't a 6) * ...

Or

P(1st throw isn't a 6) * P(2nd throw isn't a 6) * P(3d throw is a 6) * ....

In fact, there are 8 choose 3 such possibilities and you've ignored all but one of them. Each possibility has the same probability so instead of adding them all up, you can just find one possibility and then ultiply it by the number of possibilities (8 choose 3).