Question

the higher education research institute at ucla collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the u.s. 71.7% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. suppose that you randomly pick eight first-time, full-time freshmen from the survey. you are interested in the number that believes that same-sex couples should have the right to legal marital status.
construct the probability distribution function (pdf). (round your probabilities to five decimal places.)
$x$ $p(x)$
0
1
2
3
4
5
6
7
8

Explanation:

This is a binomial probability problem, where:

• Number of trials $n=8$
• Probability of success $p=0.717$
• Probability of failure $q=1-p=0.283$

The binomial probability formula is:
$$P(x) = \binom{n}{x} p^x q^{n-x}$$
where $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

Step1: Define binomial parameters

$n=8$, $p=0.717$, $q=0.283$

Step2: Calculate $P(0)$

$$P(0) = \binom{8}{0} (0.717)^0 (0.283)^8 = 1 \times 1 \times 0.283^8 \approx 0.00003$$

Step3: Calculate $P(1)$

$$P(1) = \binom{8}{1} (0.717)^1 (0.283)^7 = 8 \times 0.717 \times 0.283^7 \approx 0.00063$$

Step4: Calculate $P(2)$

$$P(2) = \binom{8}{2} (0.717)^2 (0.283)^6 = 28 \times 0.717^2 \times 0.283^6 \approx 0.00523$$

Step5: Calculate $P(3)$

$$P(3) = \binom{8}{3} (0.717)^3 (0.283)^5 = 56 \times 0.717^3 \times 0.283^5 \approx 0.02768$$

Step6: Calculate $P(4)$

$$P(4) = \binom{8}{4} (0.717)^4 (0.283)^4 = 70 \times 0.717^4 \times 0.283^4 \approx 0.09073$$

Step7: Calculate $P(5)$

$$P(5) = \binom{8}{5} (0.717)^5 (0.283)^3 = 56 \times 0.717^5 \times 0.283^3 \approx 0.19769$$

Step8: Calculate $P(6)$

$$P(6) = \binom{8}{6} (0.717)^6 (0.283)^2 = 28 \times 0.717^6 \times 0.283^2 \approx 0.28212$$

Step9: Calculate $P(7)$

$$P(7) = \binom{8}{7} (0.717)^7 (0.283)^1 = 8 \times 0.717^7 \times 0.283 \approx 0.24912$$

Step10: Calculate $P(8)$

$$P(8) = \binom{8}{8} (0.717)^8 (0.283)^0 = 1 \times 0.717^8 \times 1 \approx 0.14677$$

Answer:

$x$	$P(x)$
1	0.00063
2	0.00523
3	0.02768
4	0.09073
5	0.19769
6	0.28212
7	0.24912
8	0.14677