Each morning your alarm goes off. There is a 22% chance that you will not wake up and consequently be late for your 9am class. For the first 8 classes of the term, determine the probability that you will be ....
a) late exactly 3 times ____
b.) on time for every class ____
c) lat more than 3 times ____
d.) late at least 3 times but not more than 5 times ___
e.) suppose you were late for the first two classes. what is the probability than you will be late exactly three times?
f.) suppose you were late for the first two classes. What is the probability than you will be late more than 3 times?
Your questions fall in the category of binomial distribution
I will do the first two, you do and then show me how you did the others.
a) prob(wake up) = .78
prob(not wake up) = .22
prob(not wake up exactly 3 times out of 8)
= C(8,3) (.22^3)(.78^5)
= 56(.0030742......)
= ..17216 , rounded off to 5 decimal places
b) prob(wake up 8 out of 8 times)
= C(8,8)(.78^8)(.22^0)
= (1)(.13701)(1)
= .13701
c) late more than three times
= late 4 times + late 5 times + ...+ late 8 times
or
= 1 - (late none + late 1 + late 2 + late 3)
= ....
d) ....
To solve these probability problems, we can use the binomial probability formula. The binomial probability formula calculates the probability of getting a certain number of successes in a fixed number of independent Bernoulli trials (two possible outcomes: success or failure) with the same probability of success on each trial.
Let's define the variables:
n = number of trials
p = probability of success on each trial
a) To determine the probability of being late exactly 3 times for the first 8 classes, we have:
n = 8 (number of classes)
p = 0.22 (probability of being late)
We can use the binomial coefficient (nCr) to calculate the number of ways we can choose 3 classes out of 8:
nCr = n! / (r!(n-r)!)
Using the formula for calculating the probability:
P(X = k) = nCr * p^k * (1-p)^(n-k)
For a) P(X = 3):
P(X = 3) = 8C3 * 0.22^3 * (1-0.22)^(8-3)
b) To determine the probability of being on time for every class, we need to calculate the probability of not being late (success) for all 8 classes. Since the probability of not being late is 1 - probability of being late, we can use the formula:
P(X = k) = nCr * p^k * (1-p)^(n-k)
For b) P(X = 0):
P(X = 0) = 8C0 * 0.22^0 * (1-0.22)^(8-0)
c) To calculate the probability of being late more than 3 times, we need to find the sum of probabilities for being late 4, 5, 6, 7, or 8 times. We can use the formula:
P(X > k) = 1 - P(X <= k)
For c) P(X > 3):
P(X > 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3))
d) To calculate the probability of being late at least 3 times but not more than 5 times, we need to find the sum of probabilities for being late 3, 4, or 5 times. We can use the formula:
P(a <= X <= b) = P(X = a) + P(X = a+1) + ... + P(X = b)
For d) P(3 <= X <= 5):
P(3 <= X <= 5) = P(X = 3) + P(X = 4) + P(X = 5)
e) If you were already late for the first two classes, the remaining 6 classes will be considered as new trials. So, you need to calculate the probability of being late exactly 3 times out of the remaining 6 classes.
n = 6 (number of remaining classes)
p = 0.22 (probability of being late)
Using the formula for calculating the probability:
P(X = k) = nCr * p^k * (1-p)^(n-k)
For e) P(X = 3):
P(X = 3) = 6C3 * 0.22^3 * (1-0.22)^(6-3)
f) Similar to c), to calculate the probability of being late more than 3 times with the condition of already being late for the first two classes, you need to find the sum of probabilities for being late 1 or 2 times out of the remaining 6 classes. We can use the formula:
P(X > k) = 1 - P(X <= k)
For f) P(X > 3):
P(X > 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))
Using the binomial probability formula and the combinations formula, you can calculate the probabilities for each of the scenarios mentioned.