Question

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Consider rolling a fair die thrice and tossing a fair coin sixteen times. Assume that all the tosses and rolls are independent. and the chance that the total The chance that the total number of heads in all the coin tosses equals 12 is (Q3) number of spots showing in all the die rolls equals 12 is (Q4) The number of heads in all the tosses of the coin plus the total number of times the die lands with an even number of spots showing on top (Q5) ?

Answer 1

Solution:

Q3) Use the binomial distribution with n=16, p=1/2, k=12.

P(exactly 9 heads out of 10 tosses)
= (16 choose 12) * (1/2)^12 * (1 - 1/2)^(16 - 12)
= (1820)*(1/2)^12 * (1/2)^4
= (1820)*(1/2)^12+4
= (1820)*(1/2)^16
= (1820)(1/65536)
= 455/16384  

Q4) I'm assuming the fair die has faces numbered 1 through 6.
Note that we cannot use the binomial distribution formula, since the total number of spots on the three rolls is not a number of successes.

There are 6^3 = 216 possible equally likely outcomes for 3 rolls of the die.

We can count the number of ways of getting a total of 12, by considering each possible number on the first roll and adding the six results.

If the first die is 1, then we count the number of ways of getting a total of 11 on the other two rolls, which is 5 ways.
If the first die is 2, then we count the number of ways of getting a total of 10 on the other two rolls, which is 6 ways.
If the first die is 3, then we count the number of ways of getting a total of 9 on the other two rolls, which is 5 ways.
If the first die is 4, then we count the number of ways of getting a total of 8 on the other two rolls, which is 4 ways.
If the first die is 5, then we count the number of ways of getting a total of 7 on the other two rolls, which is 3 ways.
If the first die is 6, then we count the number of ways of getting a total of 6 on the other two rolls, which is 2 ways.

The number of ways of getting a total of 9 is 5+6+5+4+3+2 = 25.

So the probability of a total of 12 on three rolls is 25/216.

Q5) The event of getting even number of spots on a die roll is essentially no different from the event of heads on a coin toss, since the probability of an even number of spots is 3/6 = 1/2, which is also the probability of heads. So the total number of heads in all the tosses of the coin plus the total number of times the die lands with an even number of spots, has the same distribution as the total number of heads in 16+3 = 19 tosses of the coin. This distribution is binomial with n=19 and p=1/2.