A fair die is rolled 30 times. what is the expected number of sixes to turn up in 30 rolls? What is the probability of getting at least 7 sixes in 30 rolls?

Question

This is a question that is on binomial distribution. I have the formula and I also know how to enter it into my ti83 but the answer comes up differently every time. 

Answer 1

Hello, thank you for taking the time to post your question!

You’re off to a good start in terms of identifying that this is going to be a binomial question. We have n = 30 since there are 30 trials, p = 1/6 since it’s a fair die with 6 possibilities, and then you would be testing for x >= 7 in the probability computation.

Using a binomial calculator with those parameters I find the probability to be 0.2235.

For the expected value you can just multiply together n x p

E(X) = n x p = 30 x 1/6 = 5

So the expected number of sixes to turn up in 30 rolls is 5

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

Note that ₈C₅ translates to 8! divided by 5! divided by (8 − 5)!. Then ₈C₅ is [8x7x6x5x4x3x2x1] ÷ [5x4x3x2x1] ÷ [3x2x1] which will simplify to [8x7x6] ÷ [6] or 56.

For an honest six-sided die, the probability of landing a six is one out of six or 1/6.

The expected number of sixes in thirty rolls of the die is given by np or (number of rolls) times (probability of a six) or 30 × (1/6) or 5.

For probability of at least 7 sixes in 30 rolls, build the summation formula as

₃₀C₇(1/6)⁷(1-1/6)^(30-7) + ₃₀C₈(1/6)⁸(1-1/6)^(30-8) + "dot-dot-dot" + 

₃₀C₂₉(1/6)²⁹(1-1/6)^(30-29) + ₃₀C₃₀(1/6)³⁰(5/6)^(30-30).

A programmable calculator will compute this summation {written as

∑_{(n=7 to n=30)} [30!÷n!÷(30-n)!](1/6)ⁿ(5/6)^(30-n)} to a result of 0.2234628761.

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As confirmation, consider ∑_{(n=0 to n=6)} [30!÷n!÷(30-n)!](1/6)ⁿ(5/6)^(30-n)

which sums to a result of 0.7765371239.

0.2234628761 + 0.7765371239 sums to 1, which mathematically states

the dead certainty that any 30 rolls of a fair six-sided die will give

a result of either no more than six sixes or no less than seven sixes.