Question

A fair die is rolled and and n fair coins are tossed, where n is the number showing on the die. what is the probability that no heads appear?

Answer 1

The probability of no heads appearing when rolling a die and flipping the number of coins indicated is the average of the probabilities for getting no heads in 1 to 6 coin tosses, each multiplied by the probability of 1/6 for that die roll.

Step-by-step explanation:

Calculating the Probability of No Heads Appearing

To determine the probability that no heads appear when rolling a die and then flipping the number of coins shown on the die, we must first understand the probability of getting a particular number of heads from a certain number of coin tosses. Since each coin is fair, the probability of getting heads (p) or tails (q) is always 0.5. The number of coins tossed depends on the outcome of the die roll, which can be any integer from 1 to 6.

To calculate the main answer, we need to find the probability for each die outcome and then sum these probabilities. For any die roll, the probability of flipping n coins and getting no heads is (0.5)^n. We calculate this for each die face and sum the results:

1. 
   
3. For a die roll of 1: the probability of no heads is (0.5)^1
4. 
   
6. For a die roll of 2: the probability of no heads is (0.5)^2
7. 
   
9. For a die roll of 3: the probability of no heads is (0.5)^3
10. 
    
12. For a die roll of 4: the probability of no heads is (0.5)^4
13. 
    
15. For a die roll of 5: the probability of no heads is (0.5)^5
16. 
    
18. For a die roll of 6: the probability of no heads is (0.5)^6

Since each die roll is equally likely with a fair six-sided die, each outcome has a probability of 1/6. The overall probability is the average of the six probabilities calculated above, which gives our detailed answer.