Question

Among gamblers playing the slot machines and winning at a certain time​ (at a certain​ casino), the following was determined to be true about one hour​ later: 45​% were still​ winning;

25​% were​ losing; 10% had switched from the slots to table games​ (blackjack, roulette,​ etc.); and 20​% had left the casino.​ Similarly, among gamblers playing the slots and losing at a certain​ time, the following was determined to be true about one hour​ later: 32​% were​ winning; 47​% were still​ losing; 15% had switched from the slots to table​ games; and 6​% had left the casino. Among gamblers playing the slots and winning​ (at that certain​ time), what percentage eventually switch to table​ games? Round your answer to the nearest whole number.

A.

39​%

B.

45​%

C.

35​%

D.

43​%

Answer 1

We are given here that:
P(Winning --> Winning) = 0.45,
P(Winning --> Losing) = 0.25,
P(Winning --> Switched) = 0.1,
P(Winning --> Left) = 0.2

For the losing ones,
P(Losing --> Winning) = 0.32,
P( Losing --> Losing) = 0.47,
P(Losing --> Switched) = 0.15,
P(Losing --> Left) = 0.06

Let the probability of eventually leaving to switch to table games from winning be X. Also the probability of eventually leaving to switch to table games from initial losing position be Y.

Then from winning position, we have here:
X = 0.45*X + 0.25*Y + 0.1*1 + 0.2*0
0.55X = 0.25Y + 0.1

Also from losing position,
Y = 0.32*X + 0.47Y + 0.15*1 + 0.06*0
0.53Y = 0.32X + 0.15
Y = (1/0.53)*(0.32X + 0.15)

Therefore, putting this above value of Y in the previous equation, we get:
0.55X = (0.25/0.53)*(0.32X + 0.15) + 0.1
0.3991X = 0.1708
X = 0.43

Therefore 0.43 is the required probability here. Therefore 43% is the required probability here.