An airline reports that if has been experiencing a 15% rate of no-shows on advanced reservations. Among 150 advanced reservations, find the probability that there will be fewer than 20 no-shows.
To find the probability that there will be fewer than 20 no-shows, we need to use the binomial probability formula:
P(X < 20) = Σ_(k=0)^19 (nCk) * p^k * q^(n-k)
where:
P(X < 20) is the probability of having fewer than 20 no-shows,
Σ is the summation symbol,
k is the number of successful outcomes (no-shows),
n is the total number of trials (advanced reservations),
p is the probability of a successful outcome (rate of no-shows),
q is the probability of a failed outcome (1 - p),
nCk is the binomial coefficient.
Given that the rate of no-shows is 15% (0.15) and there are 150 advanced reservations, we can substitute these values into the formula:
P(X < 20) = Σ_(k=0)^19 (150Ck) * (0.15)^k * (0.85)^(150-k)
Now we can calculate this probability step-by-step:
1. Calculate the binomial coefficient for each value of k:
Starting with k = 0:
(150C0) = 1
Then for k = 1:
(150C1) = 150 / 1 = 150
Repeat this process for each value of k up to 19.
2. Calculate the probability for each value of k:
For k = 0:
(0.15)^0 * (0.85)^(150-0) = 1 * 0.85^150
Then for k = 1:
(0.15)^1 * (0.85)^(150-1) = 0.15 * 0.85^149
Repeat this process for each value of k up to 19.
3. Sum up the probabilities calculated in step 2:
P(X < 20) = Σ_(k=0)^19 [(150Ck) * (0.15)^k * (0.85)^(150-k)]
Add up the probabilities for each value of k from 0 to 19.
The resulting value will be the probability that there will be fewer than 20 no-shows among 150 advanced reservations.
To find the probability of having fewer than 20 no-shows among 150 advanced reservations, we need to use the binomial probability formula.
The binomial probability formula can be represented as:
P(X = k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of having exactly k no-shows
- n is the number of trials (advanced reservations)
- k is the number of successful outcomes (no-shows)
- p is the probability of a successful outcome (rate of no-shows)
- (n C k) is the binomial coefficient, also known as "n choose k," calculated as n! / (k! * (n-k)!), where n! is the factorial of n.
In this case, we want to find the probability of having fewer than 20 no-shows, so we need to find the cumulative probability from 0 to 19 no-shows.
P(X < 20) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 19)
Now let's calculate each individual probability.
P(X = k) = (150 C k) * (0.15^k) * (0.85^(150-k))
Using this formula, we can calculate the probabilities for each value of k from 0 to 19 and then sum them up to find P(X < 20).
Mean = np = 150 * .15 = ?
Standard deviation = √npq = √(150 * .15 * .85) = ?
Note: q = 1 - p
I'll let you finish the calculations.
Once you have the mean and standard deviation, use z-scores:
z = (x - mean)/sd
Note: x = 20
After you calculate z, check a z-table to find your probability. (Remember that the problem is asking for "fewer than 20" when you check the table.)
I hope this will help get you started.