In the game of Monopoly, when a player is in jail, they can get out of jail free
ID: 3048606 • Letter: I
Question
In the game of Monopoly, when a player is in jail, they can get out of jail free if they roll doubles"; the same faces on a pair of 6 sided dice. If they wish not taking this chance, they may pay $50 at the beginning of their turn and exit with their ensuing roll. Any player may stay in jail for at most 3 turns. If they haven't rolled doubles by their third roll, they must pay $50 and exit jail after the third roll or forfeit the game. Rolls are independent from turn to turn when you play with fair dice. What is the probability that I will roll doubles on any one roll? If Y represents the number of rolls in which doubles are rolled in three attempts, create the probability distribution for Y . What is the expected value for Y. What is the probability that I'll get out of jail free? a. b. c. d.Explanation / Answer
Answer to part a)
Total number of outcomes for 3 rolls = 6*6*6 = 216
Now total number of outcomes where we can get doubles or triples are as follows
there are 6 numbers, any one of them can be he doubles, so that fixes the two digits of doubles. The third digit is free to chose any value , so again it has 6 options out any number on the dice.
Plus since there are 3 rolls in an all , the doubles are free to take any of the three positions, Like first and second roll , or second and third roll , or first and third roll.
Thus total number of observations with doubles including triples = 6 * 6* 3 = 108
Now the triples may occur in all scenarios. 666 , 666 , 666 ..this would come three times , and this would happen with all the 6 numbers . So total of 18 numbers are being repeated three times each. We just have to keep them once
So the number of repeatetions that need to be eliminated = 18 -6 = 12
Thus total number of doubles and triples without repetition = 108-12 = 96
Hence probability of getting doubles in 3 rolls = 96 /216
Probability = 0.4444
.
Answer to part b)
The value of Y can be either 2 or 3 .The doubles can be rolled either in two rolls or 3
Out of total 96 options that get doubles , 36 of them get doubles by the second roll , and rest of them get doubles by the thrid roll
Hnece the probability distribution of Y is as follows
Y = 2 , P = 36/96 = 0.375
Y = 3 , P = (96-36)/96 = 0.625
.
Answer to part c)
Expected value of Y = sum of product of Y*Probability
Expected value of Y = 2 *0.375 +3*0.625
Expected value of Y = 2.625
.
Answer to part d)
The probability that I will get out of jail free is possible only when I get doubles by the thrid roll
The probability of getting a doubles by thrid roll = number of outcomes with doubles / total number of outcomes
Probability fo getting a doubles = 96 /216
Probability of getting out of jail for free = 96 /216 = 0.444
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