Question

A balanced Roulette wheel should come up green 1/19 of the spins. To check whether a wheel is balanced, we'll spin it 40 times, and pronounce it unbalanced if it comes up green more than 3 times. State the appropriate hypotheses. What is the probability that a fair wheel will be found to be unbalanced (you may express your answer as an unsimplified summation)? For a wheel that comes up green 1/15th of the time, what is the probability that it is found to be balanced? Again, give the summation, you need not simplify

Answer 1

The balanced roulette wheel should come up green 1/19 of the spins.

To check whether it is balanced,we'll spin it 40 times,and pronounce it unbalanced if it comes up green for more than 3 times.

The appropriate hypothesis:-

Null hypothesis:-P(green)=1/19

Alternate hypothesis:- P(green)>3/40.

To find the probability that the wheel is unbalanced.

Let,X be the random variable denoting the number of times green comes uo.

Then,X~binomial(40,1/19)

So,

P(X>3)

=P(X=4)+P(X=5)+....+P(X=39)+P(X=40)

$\sum \binom{40}{x}(1/19)^{x})(18/19)^{^{40-x}}$

where x runs from 4,5,6,....,38,39,40.

If the probability of turning up green was 1/15,

X~binomial(40,1/15)

Then the probability that the wheel was unbalanced is

P(X>3)

=P(X=4)+P(X=5)+....+P(X=39)+P(X=40)

$\sum \binom{40}{x}(1/15)^{x})(14/15)^{^{40-x}}$

where x runs from 4,5,6,....,38,39,40.