Coin flip and dice roll question (check my answers?)

In summary, the probability of getting a head on the die and the number of heads in 6 flips is 1/384.
  • #1
zeion
466
1

Homework Statement



Hi,
I just wanted to check my answers for this question:

Q: Suppose that we flip a coin six times and roll a 6-sided die once. Suppose also that all outcomes of this experiment (consisting of an ordered sequence of results for the flips (heads or tails) and the number showing on the die after the roll (an integer from 1 to 6)) are equally likely.
Find the following probabilities:
1) The probability that all six flips are heads and the die shows the number 6
2) The probability that the first R flips are heads, where R is whatever number is showing on the die.
3) The probability that the number of flips that are heads times the number showing on the die is 18.
4) The probability that the number of heads in a the six flips is same as the number showing on the die.

Homework Equations





The Attempt at a Solution



A:
1) Let A be that event that all six flips are heads.
Let B be the event that the dice roll is 6.
P(A) = #A/#S, #B = 1, #S = 2^6, so P(A) = 1/64
P(B) = #B/#S, #b = 1, #S = 6, so P(B) = 1/6
Since A and B are independent, P(A)∩P(B) = P(A)P(B) = 1/384

2) We want the roll on the die and the number of heads in 6 flips to match.
The chance to get any roll is always 1/6.
The chance to get first as head is 2^5 / 64.. first 2 as heads is 2^4/64.. etc. Each is multiplied by 1/6 (because 'and').
Let {P(A),...,P(F)} be these probabilities.
Any of them happening will suffice.
So let E be the event in the question.
P(E) = P(A) + ... + P(F) (because 'or')
= (1/6)((32+16+8+4+2+1)/64) = 63/384

3) Let E be the event.
This can only happen if we roll 3 and get 6 heads or roll 6 and get 3 heads.
Let A and B be those events, so P(E) = P(A) + P(B)
P(A) = roll 3 and 6 heads = (1/6)(1/64) = 1/384
P(B) = roll 6 and 3 heads = (1/6)(2^3/64) = 20/384
So P(E) = 21/384

4) We want number of heads and roll to be the same.
So, 1 head and roll 1 or 2 head and roll 2 or ... or 6 head and roll 6.
Rolling any number is 1/6. Permutation for 1 head any order is 6 choose 1.. 2 head is 6 choose 2 etc. All possible is 2^6 = 64.
Let E be the event, then P(E) = (1/384)(6 + 15 + 20 + 15 + 6 + 1) = 63/384

Thanks
 
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  • #2
Just one thing - for question 3 you've got 2^3 = 20. Should be 8.
 
  • #3
I think zeion just typed his work in incorrectly. Something else should be there instead of 2^3, but his final answer is correct.
 
  • #4
you're right vela - 6 choose 3 gives 20.
 
  • #5
Yes I meant 6 choose 3 thanks guys.
 

1. How does the probability of getting heads on a coin flip compare to the probability of rolling a 6 on a dice?

The probability of getting heads on a coin flip is 1/2, or 50%, while the probability of rolling a 6 on a dice is 1/6, or about 16.67%. This means that it is more likely to get heads on a coin flip than to roll a 6 on a dice.

2. What is the probability of getting tails on two consecutive coin flips?

The probability of getting tails on two consecutive coin flips is (1/2) x (1/2) = 1/4, or 25%. This is because the probability of getting tails on one flip is 1/2, and the probability of getting tails on the second flip is also 1/2. These probabilities are multiplied together to get the overall probability.

3. How many possible outcomes are there when flipping a coin and rolling a dice?

There are 12 possible outcomes when flipping a coin and rolling a dice. This is because there are two possible outcomes for the coin flip (heads or tails) and six possible outcomes for the dice roll (numbers 1-6). The total number of outcomes is found by multiplying these two numbers together: 2 x 6 = 12.

4. Is it possible to get the same outcome on both a coin flip and a dice roll?

Yes, it is possible to get the same outcome on both a coin flip and a dice roll. For example, if you get heads on the coin flip and roll a 1 on the dice, the outcome would be "heads and 1". However, this outcome is less likely to occur compared to getting different outcomes on each flip and roll.

5. What is the difference between independent and dependent events in relation to a coin flip and dice roll?

An independent event is one where the outcome of one event does not affect the outcome of another event. For example, flipping a coin and rolling a dice are independent events. A dependent event is one where the outcome of one event does affect the outcome of another event. For example, if you draw a card from a deck and then draw another card without replacing the first, the second draw is dependent on the first and the probabilities change.

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