A radio station gives a pair of concert tickets to the 6th called who knows the birthday of the performer. For each person who calls, the probability

Answer:
a)$P(L=k)=\{\begin{array}{cc}{0.25}^{(k-1)}\times 0.75\hfill & \text{for }k=1,2,3,...\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$
b)$P(L=10)={0.25}^{(10-1)}\times 0.75={0.25}^9\times 0.75$
c)$P(L\ge 9)={\sum }_{k=9}^{\infty }P(L=k)={\sum }_{k=9}^{\infty }{0.25}^{(k-1)}\times 0.75$
Explanation:
a. To find the PMF (Probability Mass Function) of L, the number of calls necessary to find the winner, we need to calculate the probability of each possible value of L.
Let's define L as the number of calls necessary to find the winner. The possible values for L are 1, 2, 3, 4, ...
The probability of finding the winner on the first call is the probability that the first caller knows the performer's birthday, which is given as 0.75.
The probability of finding the winner on the second call is the probability that the first caller doesn't know the birthday (0.25) multiplied by the probability that the second caller knows the birthday (0.75). Therefore, the probability is $P(L=2)=0.25\times 0.75$.
Similarly, the probability of finding the winner on the third call is the probability that the first two callers don't know the birthday (0.25 * 0.25) multiplied by the probability that the third caller knows the birthday (0.75). Therefore, the probability is $P(L=3)=0.25\times 0.25\times 0.75$.
In general, the probability of finding the winner on the nth call is the probability that the first (n-1) callers don't know the birthday (0.25^(n-1)) multiplied by the probability that the nth caller knows the birthday (0.75). Therefore, the probability is $P(L=n)={0.25}^{(n-1)}\times 0.75$.
Since L can take any positive integer value, the PMF is defined as:
$P(L=k)=\{\begin{array}{cc}{0.25}^{(k-1)}\times 0.75\hfill & \text{for }k=1,2,3,...\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$
b. To find the probability of finding the winner on the 10th call, we can substitute k = 10 into the PMF formula:
$P(L=10)={0.25}^{(10-1)}\times 0.75={0.25}^9\times 0.75$
c. To find the probability that the station will need 9 or more calls to find a winner, we need to sum up the probabilities of finding the winner on the 9th, 10th, 11th, 12th, ... calls. This can be calculated as follows:
$P(L\ge 9)=P(L=9)+P(L=10)+P(L=11)+...$
$P(L\ge 9)={0.25}^8\times 0.75+{0.25}^9\times 0.75+{0.25}^{10}\times 0.75+...$
In general, the probability that the station will need 9 or more calls to find a winner is:
$P(L\ge 9)={\sum }_{k=9}^{\infty }P(L=k)={\sum }_{k=9}^{\infty }{0.25}^{(k-1)}\times 0.75$

Step 1:
a. The PMF (Probability Mass Function) of L, the number of calls necessary to find the winner, can be calculated using the geometric distribution. The probability mass function is given by:
$P(L=l)=(1-p{)}^{l-1}·p$ where $p$ is the probability of knowing the performer's birthday, which is 0.75 in this case.
Step 2:
b. To find the probability of finding the winner on the 10th call, we can calculate $P(L=10)$ using the PMF formula:
$P(L=10)=(1-0.75{)}^{10-1}·0.75$
Step 3:
c. To calculate the probability that the station will need 9 or more calls to find a winner, we need to sum the probabilities of all possible values of $L$ from 9 to infinity:
$P(L\ge 9)=P(L=9)+P(L=10)+P(L=11)+\dots$
Note: In this case, since the calls are independent, the probability of finding the winner on the 10th call is the same as the probability of needing exactly 10 calls to find the winner.

a. To find the Probability Mass Function (PMF) of L, the number of calls necessary to find the winner, we can use the geometric distribution since it represents the number of trials until the first success (in this case, finding the winner).
The probability of success on each trial is given as p = 0.75 (the probability that a caller knows the performer's birthday). The probability of failure (not finding the winner) on each trial is q = 1 - p = 0.25.
The PMF of L is given by:
$P(L=k)=q^{k-1}·p$
where k is the number of trials until the first success (finding the winner).
b. The probability of finding the winner on the 10th call is:
$P(L=10)=q^{10-1}·p$
c. To find the probability that the station will need 9 or more calls to find a winner, we need to calculate the sum of probabilities for L ≥ 9:
$P(L\ge 9)=P(L=9)+P(L=10)+P(L=11)+\dots$
To calculate this, we can use the complement rule:
$P(L\ge 9)=1-P(L<9)$
$P(L<9)=P(L=1)+P(L=2)+\dots +P(L=8)$
We can substitute the values of the PMF formula to calculate these probabilities.
a. PMF of L:
$P(L=k)={0.25}^{k-1}·0.75$
b. Probability of finding the winner on the 10th call:
$P(L=10)={0.25}^{10-1}·0.75$
Substituting k = 10:
$P(L=10)={0.25}^9·0.75$
Calculating the result:
$P(L=10)=0.00095367431640625·0.75$
$P(L=10)=0.0007152557373046875$
So, the probability of finding the winner on the 10th call is approximately 0.000715 or 0.0715%.
c. Probability that the station will need 9 or more calls to find a winner:
$P(L\ge 9)=1-P(L<9)$
$P(L<9)=P(L=1)+P(L=2)+\dots +P(L=8)$
Substituting the values into the PMF formula for each k from 1 to 8 and summing them:
$P(L<9)={0.25}^0·0.75+{0.25}^1·0.75+\dots +{0.25}^7·0.75+{0.25}^8·0.75$
Calculating the result:
$P(L<9)=1·0.75+0.25·0.75+\dots +{0.25}^7·0.75+{0.25}^8·0.75$
$P(L<9)=0.75(1+0.25+{0.25}^2+\dots +{0.25}^7+{0.25}^8)$
This is a geometric series with the first term (a) equal to 1 and the common ratio (r) equal to 0.25. We can use the formula for the sum of a geometric series:
$P(L<9)=0.75·\frac{1-{0.25}^9}{1-0.25}$
Calculating the result:
$P(L<9)=0.75·\frac{1-0.00095367431640625}{1-0.25}$
$P(L<9)=0.75·\frac{1-0.00095367431640625}{0.75}$
$P(L<9)=1-0.00095367431640625$
$P(L<9)=0.99904632568359375$
Finally, we can calculate the probability that the station will need 9 or more calls to find a winner using the complement rule:
$P(L\ge 9)=1-P(L<9)$
$P(L\ge 9)=1-0.99904632568359375$
$P(L\ge 9)=0.00095367431640625$
So, the probability that the station will need 9 or more calls to find a winner is approximately 0.000954 or 0.0954%.