Question

Suppose you play 1000 spins of Roulette and on each time you bet that the ball will land on a red number

3. Use the normal approximation to compute the probability of winning

a. More than 470 times: P(X > 470)? (5-pts)

b. P (X<475)? (5-pts)

c. P(X>500)? (5-pts) i. Note winning more than 500 times is necessary to walk away a ‘winner’ if you bet the same amount each time.

d. P(X<500)? (5pts) i. Note winning less than 500 times means you walk away a ‘loser’ if you bet the same amount each time.

Answer 1

Out of 38 slots, 18 are red so

p = P(red) = 18/ 38

Since np =473.68 and n(1-p) = 526.32 both are greater than 5 so we can use normal approximation here.

Using normal approximation, X has approximately normal distribution with mean and SD as follows:

(a)

The z-score for X = 470+0.5 =470.5 is

The probability of winning more than 470 times is

P(X > 470) = P(z > -0.20) = 1 - P(z <= -0.20) = 0.5793

Answer: 0.5793

(b)

The z-score for X = 475 - 0.5 = 474.5 is

The required probability is:

P(X< 475)=P(z<0.05)=0.5199

Answer: 0.5199

(c)

The z-score for X = 500+0.5 = 500.5 is

The required probability is:

P(X > 500)=P(z > 1.70)=0.0446

Answer: 0.0446

(d)

The z-score for X = 500-0.5 =499.5 is

The required probability is:

P(X < 500)=P(z <1.64)=0.9495

Answer: 0.9495