Question

Find the indicated probability. Round to the nearest thousandth.
A study conducted at a certain college shows that 55% of the school's graduates find a job in their chosen field within a year after graduation. Find
the probability that among 7 randomly selected graduates, at least one finds a job in his or her chosen field within a year of graduating.
0.985
0.996
0.550
0.143

Answer 1

Answer:

$P(At\ least\ 1) = 0.985$

Step-by-step explanation:

Given

Proportion = 55%

Required

Probability that at least one out of 7 selected finds a job

Let the proportion of students that finds job be represented with p

$p = 55\%$

Convert to decimal

$p = 0.55$

Let the proportion of students that do not find job be represented with q

Such that;

$p + q = 1$

Make q the subject of formula

$q = 1 - p$

$q = 1 - 0.55$

$q = 0.45$

In probability; opposite probabilities add up to 1;

In this case;

Probability of none getting a job + Probability of at least 1 getting a job = 1

Represent Probability of none getting a job with P(none)

Represent Probability of at least 1 getting a job with P(At least 1)

So;

$P(none) + P(At\ least\ 1) = 1$

Solving for the probability of none getting a job using binomial expansion

$(p + q)^n = ^nC_0p^nq^0 + ^nC_1p^{n-1}q^1 +.....+^nC_np^0q^n$

Where $^nC_r = \frac{n!}{(n-r)!r!}$ and n = 7; i.e. total number of graduates

For none to get a job, means 0 graduate got a job;

So, we set r to 0 (r = 0)

The probability becomes

$P(none) = ^nC_0p^nq^0$

Substitute 7 for n

$P(none) = \frac{7!}{(7-0)!0!} * p^7 * q^0$

$P(none) = \frac{7!}{7!0!} * p^7 * q^0$

$P(none) = \frac{7!}{7! * 1} * p^7 * q^0$

$P(none) = 1 * p^7 * q^0$

Substitute $p = 0.55$ and $q = 0.45$

$P(none) = 1 * 0.55^7 * 0,45^0$

$P(none) = 0.01522435234$

Recall that

$P(none) + P(At\ least\ 1) = 1$

Substitute $P(none) = 0.01522435234$

$0.01522435234+ P(At\ least\ 1) = 1$

Make P(At least 1) the subject of formula

$P(At\ least\ 1) = 1 - 0.01522435234$

$P(At\ least\ 1) = 0.98477564766$

$P(At\ least\ 1) = 0.985$ (Approximated)