Probability - Coin Toss - Find Formula

In summary: P(A|B) = (n choose k)/2^n / (n+1 choose k)/2^n = (n choose k)/(n+1 choose k) = k/(n+1)In summary, we have found the simple formulae for both parts of the problem:a) P(k-1 heads |k-1 or k heads) = k/(n+1)b) P(k heads |k-1 or k heads) = k/(n+1)Keep up the good work with your studies and best of luck!
  • #1
bizboy1
11
0

Homework Statement


Suppose a fair coin is tossed n times. Find simple formulae in terms of n and k for

a) P(k-1 heads |k-1 or k heads)
b) P(k heads | k-1 or k heads)


Homework Equations


P(k heads in n fair tosses)=binomial(n,k)/2^n (0<=k<=n)


The Attempt at a Solution


I'm stuck on the conditional probability. I've dabbled with it a little bit but I'm confused what k-1 intersect k is. This is for review and not homework. The answer to a) is k/(n+1).
I tried P(k-1 heads | k heads)=P(k-1 intersect K)/P(K heads)=P(K-1)/P(K). I also was thinking about P(A|A,B)=P(An(AuB))/P(AuB)=P(Au(AnB))/P(AuB)=P(A)/(P(A)+P(B)-P(AB))

Thanks!
 
Physics news on Phys.org
  • #2


Thank you for your question. It seems like you are on the right track with your attempts at solving the problem. Here are some steps you can follow to find the simple formulae for both parts of the problem:

a) P(k-1 heads |k-1 or k heads)

First, let's define the events involved in this conditional probability:

A: k-1 heads in n tosses
B: k-1 or k heads in n tosses

Using the definition of conditional probability, we can rewrite the expression as:

P(k-1 heads |k-1 or k heads) = P(A|B) = P(A and B)/P(B)

Now, let's calculate the probabilities involved:

P(A and B) = P(k-1 heads and k-1 or k heads) = P(k-1 heads) = (n choose k-1)/2^n

P(B) = P(k-1 or k heads) = P(k-1 heads) + P(k heads) = (n choose k-1)/2^n + (n choose k)/2^n = (n+1 choose k)/2^n

Substituting these values back into our expression, we get:

P(k-1 heads |k-1 or k heads) = P(A|B) = (n choose k-1)/2^n / (n+1 choose k)/2^n = (n choose k-1)/(n+1 choose k) = k/(n+1)

b) P(k heads |k-1 or k heads)

Following a similar approach as above, we can define the events involved as:

A: k heads in n tosses
B: k-1 or k heads in n tosses

And rewrite the expression as:

P(k heads |k-1 or k heads) = P(A|B) = P(A and B)/P(B)

Calculating the probabilities:

P(A and B) = P(k heads and k-1 or k heads) = P(k heads) = (n choose k)/2^n

P(B) = P(k-1 or k heads) = P(k-1 heads) + P(k heads) = (n choose k-1)/2^n + (n choose k)/2^n = (n+1 choose k)/2^n

Substituting these values back into our expression, we get:

P(k
 

What is the probability of getting heads when tossing a coin?

The probability of getting heads when tossing a coin is 50%. This means that out of every 2 coin tosses, you can expect to get heads once.

What is the formula for calculating the probability of getting a specific outcome in a coin toss?

The formula for calculating the probability of getting a specific outcome in a coin toss is P = Number of desired outcomes / Total number of possible outcomes. For example, if you want to calculate the probability of getting heads, the formula would be P = 1/2 = 0.50.

What are the possible outcomes when tossing a coin?

The possible outcomes when tossing a coin are heads and tails. These two outcomes are equally likely to occur and have a 50% chance of happening.

How does the number of coin tosses affect the probability of getting a specific outcome?

The number of coin tosses does not affect the probability of getting a specific outcome. Each coin toss is an independent event and the probability remains the same regardless of the number of tosses.

What is the difference between theoretical probability and experimental probability?

Theoretical probability is based on mathematical calculations and predicts the likelihood of an event occurring. Experimental probability is based on actual data collected from experiments and is an estimation of the likelihood of an event occurring.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
625
  • Calculus and Beyond Homework Help
Replies
1
Views
253
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
2
Replies
37
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
759
  • Calculus and Beyond Homework Help
Replies
6
Views
813
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
Back
Top