n people put their phones into a box
Kiera has a white phone.
Assume each of the remaining n-1 phones has probability p of being white.
Kiera takes a phone back out of the basket.
Given that the phone she picks is white, what is the probability that the phone she picked is actually hers?
To calculate the probability that the phone Kiera picked is actually hers, we will use Bayes' theorem.
Let's denote the event that Kiera picks her own phone as A, and the event that the phone she picks is white as B. We want to find P(A|B), the probability that Kiera's phone is picked given that it is white.
According to Bayes' theorem, P(A|B) = (P(B|A) * P(A)) / P(B)
First, let's calculate P(B|A), the probability that Kiera picks a white phone given that it is her own phone. Since Kiera has a white phone, P(B|A) = 1.
Next, let's calculate P(A), the probability that Kiera's phone is picked. Since there are n phones in total, the probability of Kiera's phone being picked is 1/n.
Now, let's calculate P(B), the probability of picking a white phone. This is equal to P(B|A) * P(A) + P(B|not A) * P(not A). The probability of picking a white phone given that it is not Kiera's phone (not A) is (n-1)/n, since there are (n-1) remaining phones that are not hers. The probability of not picking Kiera's phone, P(not A), is (n-1)/n.
Therefore, P(B) = 1 * (1/n) + ((n-1)/n) * ((n-1)/n)
Now we can substitute these values into the equation for Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
= (1 * (1/n)) / (1 * (1/n) + ((n-1)/n) * ((n-1)/n))
Simplifying this expression, we get:
P(A|B) = 1 / (1 + (n-1)^2/n^2)
To find the probability that the phone Kiera picked is actually hers, given that it is white, we can use Bayes' theorem.
Let's define the events:
A = Phone picked by Kiera is hers
B = Phone picked by Kiera is white
We want to find P(A|B), the probability that Kiera's phone is picked given that it is white.
Bayes' theorem states:
P(A|B) = (P(B|A) * P(A))/P(B)
1. P(B|A): The probability that the phone picked by Kiera is white, given that it is hers. Since Kiera's phone is white, this probability is 1.
2. P(A): The probability that Kiera's phone is picked. Out of n phones, the probability of choosing Kiera's phone is 1/n.
3. P(B): The probability that any phone picked is white. This can be calculated by considering two cases:
a. Kiera's phone is picked: The probability of picking Kiera's phone and it being white is (1/n) * 1 = 1/n.
b. Any other phone is picked: The probability of picking any other phone and it being white is ((n-1)/n) * p, where p is the probability that any other phone is white.
Therefore, P(B) = (1/n) + ((n-1)/n) * p.
Substituting these values into Bayes' theorem, we get:
P(A|B) = (1 * (1/n))/((1/n) + ((n-1)/n) * p)
= 1/(1 + (n-1)p)
Therefore, the probability that the phone Kiera picked is actually hers, given that it is white, is 1/(1 + (n-1)p).