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

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Homework Help Overview

The problem involves rolling a fair die and subsequently tossing a coin a number of times equal to the die's outcome. The focus is on determining the probability mass function for the number of heads obtained from the coin tosses.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the values that the random variable representing the number of heads can take and question how to proceed with the calculations. Some suggest direct computation of probabilities based on the outcomes of the die roll.

Discussion Status

There are various attempts to break down the problem, with some participants providing insights into the probabilities associated with rolling different numbers on the die and the corresponding coin flips. Multiple interpretations of the probability mass function are being explored, but there is no explicit consensus on the approach to take.

Contextual Notes

Participants note the uniform probability of rolling each number on the die and the probability of heads when flipping the coin. There is some confusion regarding how to combine these probabilities effectively.

kingwinner
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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!
 
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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.
 
By the way, is this a "birvariate" distirbution? How can I define the other random variable?
 
Could somebody please help? I am totally stuck...
 
Start with the probability of rolling a one...what is that?
 
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
 
gabbagabbahey said:
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...
 
Probability of 6 heads= 1/6 times (1/2)^6

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

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