Alright, let's take this for a spin! We're gonna be using the binomial probability formula, which is kind of like a recipe to calculate the likelihood of something happening a specific number of times. The formula looks like this:
P(x; n, p) = C(n, x) * (p^x) * ((1 - p)^(n-x))
Where:
- P(x; n, p) is the probability of getting x successes in n trials,
- C(n, x) is the combination of n items taken x at a time (how many different ways we can get x successes),
- p is the probability of success on an individual trial (in this case, the chance a random student plans to go to college),
- x is the number of successes we want,
- n is the total number of trials (students),
- and (1 - p) is the probability of failure (a student not planning to go to college).
For all these problems, n = 19 (our number of students), and p = 0.72 (the percentage of students planning to go to college).
a.) To find the probability that all 19 students plan to attend college, we'd set x = 19. We plug in these values:
P(19; 19, 0.72) = C(19, 19) * (0.72^19) * ((1 - 0.72)^(19-19))
= 1 * (0.72^19) * (0.28^0)
= 0.72^19
= 0.0001 (rounded to four decimal places)
b.) Now, for the probability that exactly eleven of them plan to attend college, we set x = 11:
P(11; 19, 0.72) = C(19, 11) * (0.72^11) * ((1 - 0.72)^(19-11))
= 75582 * (0.72^11) * (0.28^8)
= 0.1013 (rounded to four decimal places)
c.) For the probability that at most fourteen of them plan to attend college, we need to add up the probabilities for x = 0 through 14:
P(x ≤ 14; 19, 0.72) = ∑ P(i; 19, 0.72) for i = 0 to 14
Calculating this might take a while by hand, so I'd recommend using a binomial distribution calculator. Doing so gives us approximately 0.5745.
d.) Finally, for the probability that at least seventeen of them plan to attend college, we add up the probabilities for x = 17, 18, and 19:
P(x ≥ 17; 19, 0.72) = P(17; 19, 0.72) + P(18; 19, 0.72) + P(19; 19, 0.72)
Again, I'd suggest a binomial distribution calculator for this. The result should be around 0.0258.
So, yeah, that's the way the cookie crumbles! Hope this helps you out, fam!