What Is the Probability Mass Function When Rolling a Die and Tossing a Coin?

[kingwinner]

Homework Statement

Suppose we roll a fair die. Whatever number comes up, we toss a coin that many times. What is the probability mass function of the number of heads?

Homework Equations

N/A

The Attempt at a Solution

Let X=number of heads
Then I know that X can take on the values 0,1,2,3,4,5,6.
How to proceed from here I have no clue...


Could someone please explain? Any help would be appreciated!

 

[HallsofIvy]

Just direct computation. The probability of 0 is 0 because there is no "0" on a die. What is the probability of rolling a 1 on a die? Of a 2? etc.

 

[kingwinner]

By the way, is this a "birvariate" distirbution? How can I define the other random variable?

 

[kingwinner]

Could somebody please help? I am totally stuck...

 

[gabbagabbahey]

Start with the probability of rolling a one...what is that?

 

[Office_Shredder]

Literally, see what the probability of getting 1 heads is by finding the probability you roll x on the die, and multiplying by the probability of getting 1 heads in x coin flips. Rinse and repeat

 

[kingwinner]

Start with the probability of rolling a one...what is that?

Uniform prob. of 1/6 for each number on the die
And the prob. of a head is 1/2
...but I am still having trouble combining these...

 

[borgwal]

Probability of 6 heads= 1/6 times (1/2)^6

Probabilty of 5 heads=...(no, it's not just 1/6 times (1/2)^5)

 

[HallsofIvy]

Take it ones step at a time.

There is a 1/6 probability of rolling a "1" and then you flip the coin once which has probability 1/2 of getting a head: probability of getting 1 head this way is (1/6)(1/2)= 1/12.

There is a probability of 1/6 of rolling a 2 and then you flip the coin twice which has a probability 1/2 go getting one head and a probability 1/4 of getting two heads.
Probability of getting 1 head this way is 1/12 and probability of getting two heads is 1/24.

Do you see a pattern? If you roll an "n" on the coin you flip the coin n times. You have a probability of 1-(1/2)^n of getting a 1. Altogether the probability getting a one is (1/6)(1/2+ 1/4+ 1/8+ ...). What is the probability of getting two heads if you fllip a coin n times? What is the probability of rolling an "n"?