Question

In a game show, a wheel of fortune is rotated 3 times with three sectors of the same size in the colors blue, red and yellow. What is the probability of the following events?

a.) The wheel does not stop in any of the 3 attempts in the red sector. ´=

b.) The blue sector is hit exactly twice. =

c.) The sector with the color yellow appears at least twice.=

You can enter your results in the form of commas (rounded to 3 digits)

On a cloudless August evening, an average of 4 shooting stars per hour are observed.

It can be assumed that the number of shooting stars observed in t minutes is Poisson-distributed.

(a) Consider the conditions under which the number of observed shooting stars in t minutes can be modeled as a Poisson process.

(b) What is the probability of observing at least two shooting stars during half an hour?

Round the probability to 2 decimal places.

(c) How many shooting stars do you expect to see the cloudless sky for 3 consecutive nights of 6 hours each?

Answer 1

Here the wheel of fortune is rotated 3 times with three sectors of the same size in the colors blue, red and yellow.

Therefore the probability of the wheel stop any one of the sector is same and it is equal to 1/3

a) Here we want to find the probability of the wheel does not stop in any of the 3 attempts in the red sector.

That means each time it stop either blue or yellow sector

so the required probability = (2/3)^3 = 8/27 = 0.296296

b) P( the blue sector is hit exactly twice ) = P(BBR)+P(BRB)+P(RBB)+P(BBY)+P(BYB)+P(YBB) = 6/27 = 2/9 = 0.222222

Here we can use binomial with n = 3 and p = 1/3

P( the blue sector is hit exactly twice ) = P( X = 2) = "=BINOMDIST(2,3,(1/3),0)" = 0.222222

c) P(the color yellow appears at least twice) = P( X >= 2) = P( X = 2) + P(X = 3) = 2/9 + 1/27 = 7/27 = 0.259259