Suppose the number of cell phones in a household has a binomial distribution with parameters n=24 and p=25%.
Find the probability of a household having:
(a) 11 or 16 cell phones
(b) 14 or fewer cell phones
(c) 18 or more cell phones
(d) fewer than 16 cell phones
(e) more than 14 cell phones
same type of question as the previous one, except even more
tedious button pushing.
e.g.
c) 18 or more
= C(24,18) (.25^18) (.75^6) + C(24,19)(.25^19)(.75^5) + ....
.... + C(24,24)(.25^24)(.75^0)
= ....
good grief!, hope you have a good calculator
To calculate the probabilities for the given scenarios, we will use the binomial probability formula:
P(X = k) = (nCk) * (p^k) * (1-p)^(n-k)
Where:
P(X = k) is the probability of having exactly k successes (cell phones in this case)
n is the number of trials (households in this case)
p is the probability of success (probability of having a cell phone in this case)
k is the number of successes (number of cell phones in this case)
nCk is the binomial coefficient, calculated as n!/((n-k)! * k!)
Now let's solve each scenario step-by-step:
(a) Probability of having 11 or 16 cell phones:
P(X = 11) = (24C11) * (0.25^11) * (0.75^13)
P(X = 16) = (24C16) * (0.25^16) * (0.75^8)
P(a) = P(X = 11) + P(X = 16)
(b) Probability of having 14 or fewer cell phones:
P(X ≤ 14) = P(X = 0) + P(X = 1) + ... + P(X = 14)
We can calculate each individual probability and sum them up.
(c) Probability of having 18 or more cell phones:
P(X ≥ 18) = 1 - P(X < 18)
To calculate P(X < 18), we can use the complement rule:
P(X < 18) = 1 - P(X ≥ 18)
We will calculate P(X ≥ 18) as shown in (d) and subtract it from 1.
(d) Probability of having fewer than 16 cell phones:
P(X < 16) = P(X = 0) + P(X = 1) + ... + P(X = 15)
We can calculate each individual probability and sum them up.
(e) Probability of having more than 14 cell phones:
P(X > 14) = 1 - P(X ≤ 14)
We will calculate P(X ≤ 14) as shown in (b) and subtract it from 1.
Now you can use the formulas and calculations mentioned above to find the desired probabilities for the given scenarios.
To find the probabilities in this binomial distribution, we can use the binomial probability formula:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
where n is the number of trials, p is the probability of success in each trial, X is the random variable representing the number of successes, k represents a specific number of successes, and (nCk) is the binomial coefficient, which can be calculated as n! / (k! * (n-k)!)
Now, let's calculate the probabilities:
(a) To find the probability of a household having 11 or 16 cell phones, we need to find the probabilities for each individual case and add them together.
P(X=11) = (24C11) * (0.25^11) * (0.75^13)
P(X=16) = (24C16) * (0.25^16) * (0.75^8)
P(a) = P(X=11) + P(X=16)
(b) To find the probability of a household having 14 or fewer cell phones, we need to calculate the probabilities for each individual case and add them together.
P(X=0) + P(X=1) + P(X=2) + ... + P(X=14)
(c) To find the probability of a household having 18 or more cell phones, we need to calculate the probabilities for each individual case from 18 to 24 and add them together.
P(X=18) + P(X=19) + P(X=20) + ... + P(X=24)
(d) To find the probability of a household having fewer than 16 cell phones, we can subtract the probability of having 16 or more cell phones from 1.
P(X<16) = 1 - P(X>=16)
P(X>=16) = P(X=16) + P(X=17) + P(X=18) + ... + P(X=24)
(e) To find the probability of a household having more than 14 cell phones, we can subtract the probability of having 14 or fewer cell phones from 1.
P(X>14) = 1 - P(X<=14)
P(X<=14) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=14)
To calculate the individual probabilities, we can substitute the values of n, p, and k into the binomial formula and perform the calculations.