A biased coin is tossed 5 times where p(t) = .6. determine the probability that if you have 2 tails, you have 3 tails.
It is a binomial distribution, with
p=0.6 (probability of success)
q=0.4 (probability of failure)
n=5 (number of steps)
r=2 or 3 (number of successes)
P(n,r)=nCr p^r q^(n-r)
P(5,2)=5C2 0.6^2 0.4^3
=(5!)/(3!2!) 0.6^2 0.4^3
= 10(0.36)(0.064)
= 0.2304
Similarly, for 3 tails, r=3,
P(5,3)=.3456
thanks so much!
To find the probability that if you have 2 tails, you have 3 tails, we need to use conditional probability. Let's break down the problem step by step:
Step 1: Determine the probability of getting 2 tails in 5 tosses.
The probability of getting 2 tails in 5 tosses can be calculated using the binomial probability formula:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
In this case, n (the number of tosses) is 5, and p (the probability of getting a tail) is 0.5.
P(X=2) = (5C2) * 0.5^2 * (1-0.5)^(5-2)
= (10) * 0.25 * 0.125
= 0.3125
So, the probability of getting 2 tails in 5 tosses is 0.3125.
Step 2: Determine the probability of getting 3 tails in 5 tosses, given that there are 2 tails.
To find this probability, we will use conditional probability:
P(A|B) = P(A and B) / P(B)
Where:
P(A|B) is the probability of event A happening given that event B has occurred.
P(A and B) is the probability of both events A and B happening.
P(B) is the probability of event B happening.
In this case:
Event A: Getting 3 tails in 5 tosses.
Event B: Getting 2 tails in 5 tosses.
P(A and B) = P(A) * P(B|A)
P(A) is the probability of getting 3 tails in 5 tosses, which can be calculated using the same binomial formula as before.
P(A) = P(X=3) = (5C3) * 0.5^3 * (1-0.5)^(5-3)
= (10) * 0.125 * 0.125
= 0.15625
P(B|A) is the conditional probability of getting 2 tails in 5 tosses, given that we already have 3 tails. Since we already have 3 tails, we only have 2 remaining tosses, so the probability can again be calculated using the binomial formula.
P(B|A) = P(X=2) = (2C2) * 0.5^2 * (1-0.5)^(2-2)
= 1 * 0.25 * 1
= 0.25
Now, let's calculate P(A and B):
P(A and B) = P(A) * P(B|A)
= 0.15625 * 0.25
= 0.0390625
Step 3: Calculate P(B), the probability of getting 2 tails in 5 tosses (which we already calculated in Step 1):
P(B) = 0.3125
Step 4: Calculate P(A|B), the probability of getting 3 tails in 5 tosses given that there are 2 tails:
P(A|B) = P(A and B) / P(B)
= 0.0390625 / 0.3125
= 0.125
Therefore, the probability that if you have 2 tails, you have 3 tails is 0.125, or 12.5%.