Question

A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black.The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. (a)If you bet on “red,” you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette? (b)You decide to play roulette four times, each time betting on red. What is the distribution of X, the number of times you win? (c)If you bet the same amount on each play and win on exactly four of the eight plays, then you will “break even.” What is the probability that you will break even? (d)If you win on fewer than four of the eight plays, then you will lose money. What is the probability that you will lose money?

Answer 1

From the Given Problem , we have ,

The total no. of slots in the wheel = 38

The no. of Green slots = 2

The no. of Black slots = 18

The no. of Red slots = 18

a) The Probability of Winning with a bet on Red in a single play of roulette :

   

since, Here Red slots = 18 ,

Now probability rounded to 4 decimals

Probability (R) 0.4737

b) If roulette is played 4 times and each time letting on red then the distribution follows binomial distribution .

Mean = np

= 4 * 0.4737   (since p = 0.4737 )

Mean = 1.8948

SD =

= 0.998616

There fore the distribution of X follows Binomial distribution with ( n= 4 , p = 0.4737) .

C)

Assume, X denote the no .of wins.

X follows Binomial distribution with (n=4, p=0.4737)

the probability that break even when each play and win on exactly two of the four plays,

= 0.372928

P( X=2) 0.3729

D)

  The probability that you will lose money if you win less than two plays out of four.

Therefore,

P(will lose money) = P(X <2)

=P(x=0) + P(x=1)

= 0.076724 + 0.276225

= 0.35295

P ( will lose money )

   Therefore, The probability that you will lose money is 0.3530.