R. H. Bruskin Associates Market Research found that 40% of Americans do not think that having a
college education is important to succeed in the business world. If a random sample of five Americans
is s selected, find these probabilities.
(a) Exactly two people will agree with that statement.
(b) At most three people will agree with that statement.
(c) At least two people will agree with that statement.
(d) Fewer than three people agree with that statement.
Calculate the probabilities that there will be 0,1,2,3,4 and 5 persons that agree.
P(5) = (0.4)^5*[5!/(0!*5!]= 0.4^5
= 0.01024
P(0) = (0.6)^5*[5!/(0!*5!]= (0.6)^5
= 0.07776
P(1) = (0.4)*(0.6)^4*[5!/(4!*1!)]
= 0.25920
P(2) = (0.4)^2*(0.6)^3*[5!/(3!*2!)]
= 0.34560
etc. Calculate P(3) and P(4) using the same formula, and use the results to answer the questions for (a) through (d).
For (a), the answer is P(2) = 0.3456
For (d), the answer is P(0) + P(1) + P(2) = 0.68256
Well, let's put our funny hats on and calculate these probabilities, shall we?
(a) Exactly two people will agree with that statement.
To find this probability, we need to calculate the probability of two people agreeing and three people disagreeing. The probability of one person agreeing is 40%, so the probability of two people agreeing is (0.4)^2 = 0.16. The probability of three people disagreeing is (0.6)^3 = 0.216. Now, we multiply these two probabilities together:
P(exactly two people agree) = 0.16 * 0.216 ≈ 0.0346.
(b) At most three people will agree with that statement.
To find this probability, we need to calculate the probabilities of zero, one, two, and three people agreeing. We've already calculated the probability of two people agreeing in part (a), so we'll use that value here. For zero people agreeing, the probability is (0.6)^5 = 0.07776. For one person agreeing, the probability is 5 * 0.4 * (0.6)^4 ≈ 0.2304. For three people agreeing, the probability is 5 * (0.4)^3 * (0.6)^2 ≈ 0.3456. Now, we sum up these probabilities:
P(at most three people agree) = P(0) + P(1) + P(2) + P(3) ≈ 0.07776 + 0.2304 + 0.0346 + 0.3456 ≈ 0.6884.
(c) At least two people will agree with that statement.
To find this probability, we need to calculate the probabilities of two, three, four, and five people agreeing. We have already calculated the probability of two people agreeing in part (a), so we'll use that value here. The probability of three people agreeing is 5 * (0.4)^3 * (0.6)^2 ≈ 0.3456. The probability of four people agreeing is 5 * (0.4)^4 * (0.6) ≈ 0.1152. The probability of five people agreeing is (0.4)^5 ≈ 0.01024. Now, we sum up these probabilities:
P(at least two people agree) = P(2) + P(3) + P(4) + P(5) ≈ 0.0346 + 0.3456 + 0.1152 + 0.01024 ≈ 0.50564.
(d) Fewer than three people agree with that statement.
To find this probability, we need to calculate the probabilities of zero, one, and two people agreeing. We have already calculated the probability of two people agreeing in part (a), so we'll use that value here. The probability of zero people agreeing is (0.6)^5 ≈ 0.07776. The probability of one person agreeing is 5 * 0.4 * (0.6)^4 ≈ 0.2304. The probability of two people agreeing is 0.0346 (calculated in part (a)). Now, we sum up these probabilities:
P(fewer than three people agree) = P(0) + P(1) + P(2) ≈ 0.07776 + 0.2304 + 0.0346 ≈ 0.34276.
So there you have it! The probabilities of different outcomes based on the given information. I hope these probabilities brought a smile to your face!
To solve these probability problems, we will use the binomial distribution formula. The binomial distribution is used to calculate the probability of a specific number of successes in a fixed number of independent Bernoulli trials (success/failure experiments) with the same probability of success.
In this case, the success is defined as an American who does not think that having a college education is important to succeed in the business world. The probability of success is given as 40% or 0.4, and the sample size is 5 Americans.
The binomial distribution formula is:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) represents the probability of exactly k successes
- (n C k) represents the number of combinations of n items taken k at a time
- p is the probability of success
- n is the sample size
- k is the number of successes
Let's calculate the probabilities for each question:
(a) Exactly two people will agree with that statement.
Using the binomial distribution formula, the probability is:
P(X=2) = (5 C 2) * (0.4)^2 * (1-0.4)^(5-2)
P(X=2) = (5 C 2) * 0.4^2 * 0.6^3
P(X=2) = 10 * 0.16 * 0.216
P(X=2) ≈ 0.3456 or 34.56%
(b) At most three people will agree with that statement.
P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)
P(X ≤ 3) = (5 C 0) * 0.4^0 * 0.6^5 + (5 C 1) * 0.4^1 * 0.6^4 + (5 C 2) * 0.4^2 * 0.6^3 + (5 C 3) * 0.4^3 * 0.6^2
P(X ≤ 3) = (1) * 1 * 0.07776 + (5) * 0.4 * 0.1296 + (10) * 0.16 * 0.216 + (10) * 0.064 * 0.36
P(X ≤ 3) ≈ 0.98304 or 98.304%
(c) At least two people will agree with that statement.
To find the probability of "at least" something, we can calculate the complement of the event "less than." So, P(at least 2) = 1 - P(fewer than 2).
P(at least 2) = 1 - P(X=0) - P(X=1)
P(at least 2) = 1 - (5 C 0) * 0.4^0 * 0.6^5 - (5 C 1) * 0.4^1 * 0.6^4
P(at least 2) = 1 - 1 * 1 * 0.07776 - 5 * 0.4 * 0.1296
P(at least 2) ≈ 0.99264 or 99.264%
(d) Fewer than three people agree with that statement.
This is the same as P(X < 3) = P(X=0) + P(X=1) + P(X=2).
P(X < 3) = P(X=0) + P(X=1) + P(X=2)
P(X < 3) = (5 C 0) * 0.4^0 * 0.6^5 + (5 C 1) * 0.4^1 * 0.6^4 + (5 C 2) * 0.4^2 * 0.6^3
P(X < 3) = 1 * 1 * 0.07776 + 5 * 0.4 * 0.1296 + 10 * 0.16 * 0.216
P(X < 3) ≈ 0.34560 or 34.560%
there are 4 dimes, 4 nickels,
and 2 quarters.
In how many possible ways can the selection be made so that
the value of the coins is at least 25 cents?
I= important
N=not important
P(i) = 60/100 = 3/5
P(N) = 2/5
a) P(2out of5 for I) = C(5,2)(3/5)^2 (2/5)^3 = 10(9/25)(8/125) = 144/625 or .2304
..
d) fewer than three agree --> 0outof5 or 1outof5 or 2outof5 will agree
= C(5,0)(3/5)^0 (2/5)^5 + C(5,1)(3/5)^1 (2/5)^4 + C(5,2)(3/5)^2 (2/5)^3
= 1(1)(32/3125 + 5(3/5)(16/625) + 10(9/25)(8/125)
= 32/3125 + 48/625 + 144/625
= 992/3125 or .31744
do b) and c) the same way