Conditional Probability Coin Flipping Question

In summary, the experiment involves flipping a single coin with probability p of heads, and determining the number of heads obtained in three steps. The question is asking for the probability of getting 0 heads in the third step, given that the first step resulted in 1 head. This can be solved by creating a tree of possibilities and calculating the probabilities at each step.
  • #1
Yagoda
46
0

Homework Statement


The following experiment involves a single coin with probability p of heads on anyone flip, where
0 < p < 1.

Step 1: Flip the coin. Let X = 1 if heads, 0 otherwise.
Step 2: Flip the coin (X + 1) times. Let Y = the number of heads obtained in this step.
Step 3: Flip the coin (X + Y + 1) times. Let Z = the number of heads obtained in this step.
Let T denote the total number of heads across all three steps.

What is P(X = 1|Z = 0)?


Homework Equations


[itex]P(A|B) = \frac{P(A \cap B)}{P(B)}[/itex]



The Attempt at a Solution


I think I have been thinking about this too long and am just confusing myself. My first gut reaction was to say that no matter what the outcome of Z, since the coin isn't changing, the probability of it coming up heads on any given flip (ie P(X = 1)) will be p.
But since you are flipping a variable number of times to get Z, it seems your chance of getting Z = 0 would be greater with a smaller number of flips, which would be more likely if you begin with X=0 than X=1. Does this even matter?
I tried using the above conditional probability formula as well, but it got ugly quickly in trying to calculate the numerator. Is there a less thorny method that I'm missing?
 
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  • #2
Have you throught about doing a tree of the possibilities?
 
  • #3
I did try one, but I was having trouble keeping track of all the relevant numbers: possible flips, number of heads, corresponding probabilities that I was hoping there might be a clearer way. I might have to give it another crack...
 
  • #4
Yagoda said:
I did try one, but I was having trouble keeping track of all the relevant numbers: possible flips, number of heads, corresponding probabilities that I was hoping there might be a clearer way. I might have to give it another crack...

You MUST keep track of the possible numbers of heads, etc., whether it is troublesome or not. And yes, it might be lengthy and require quite a bit of work, but that is the most straightforward way to solve the problem.
 

1. What is conditional probability in the context of coin flipping?

Conditional probability in the context of coin flipping refers to the likelihood of a certain outcome occurring given that another event has already happened. For example, if we have flipped a coin and it has landed on heads, the conditional probability of getting heads again on the next flip would be different than if we had gotten tails on the first flip.

2. How is conditional probability calculated for coin flipping?

Conditional probability is calculated by dividing the probability of the two events occurring together by the probability of the first event occurring. In the case of coin flipping, this would mean dividing the probability of getting heads on both flips by the probability of getting heads on the first flip.

3. Can you provide an example of conditional probability in coin flipping?

One example of conditional probability in coin flipping is the Monty Hall problem. In this scenario, there are three doors and behind one door is a car and behind the other two are goats. The player chooses a door and then the host, who knows which door has the car, opens one of the other doors to reveal a goat. The player is then given the option to switch their choice to the remaining unopened door. The conditional probability in this situation is that the player's initial choice was incorrect and that the car is behind the other unopened door.

4. How does the number of flips affect conditional probability in coin flipping?

The number of flips does not directly affect conditional probability in coin flipping. However, as the number of flips increases, the probability of certain outcomes may change due to the law of large numbers. For example, in a series of 10 coin flips, the probability of getting heads on each flip is 50%. But in a series of 1000 coin flips, the probability of getting heads on each flip may be closer to 49.9% due to the law of large numbers.

5. What is the relationship between independent and conditional probability in coin flipping?

In coin flipping, the probability of an event occurring on one flip is independent of the probability of the same event occurring on a subsequent flip. However, conditional probability takes into account the outcome of previous flips and can affect the likelihood of certain outcomes on future flips. For example, if we know that the first two flips were both heads, the conditional probability of getting heads on the third flip would be 50% rather than 25%.

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