Question

dr. hong dai
9/23/2025
midterm exam
please submit your answers on \mc sheet of m204 midterm exam\ on canvas before 11:59 on 9/23, tuesday.
multiple choice. choose the one alternative that best completes the statement or answers the question.

1. a student asked dr. dais birthday at randomly. find the probability that dr. dais birthday is in august. (ignore leap years), pdr. dais bd is in august =

a) 1/12 b) 28/365 c) 30/365 d) 31/365
find the indicated probability.

1. given sample space s ={e1, e2, e3} with p(e1)=0.45, p(e2)=0.23, what is the subjective probability of e3, p(e3)= _?

a) 0.30 b) 0.77 c) 0.32 d) 0.55
find the empirical probability of the event.

1. the results of a survey on \whom, trump (t) or biden (b), do you think a winner of the usa 2020 presidential debate?\ were t, b, b, t, b, t, b, b, b, t, t, b, t, b, t, b. estimate the empirical probability that \trump will win\, ptrump will be president =

a) 56.25% b) 43.75% c) 62.50% d) 37.50%

1. in problemes 4 through 11, given information: in an insurance company, of 68 employees, 21 have degrees in math (m), 33 have degrees in economics (e), x have degrees in both and 26 have none of degrees in math and economics. one employee was interviewed. find the probability that the employee has a degree in both math and economics, pm ∩ e = _ hint: find x = _ first

a) 27.94% b) 23.53% c) 17.65% d) 14.71%

1. find p(e ∩ m)^c = _ by complement law.

a) 14/17 b) 7/34 c) 13/29 d) 15/19

1. find the probability that the employee has a degree in math, pm =

a) 0.2104 b) 0.2532 c) 0.2811 d) 0.3088

1. find the probability that the employee has a degree in either math or economics, pm ∪ e =

a) 12/68 b) 61/68 c) 54/68 d) 42/68

1. what is the probability that the employee has math degree if known the employ had economic gegree, pm | e = _?

a) 33/68 b) 25/42 c) 14/33 d) 4/11

1. what is the probability that the employee has an economic degree if known the employ had math gegree, pe | m = _?

a) 5/9 b) 4/7 c) 3/5 d) 2/3

Explanation:

Step1: Calculate probability of birthday in August

There are 31 days in August and 365 days in a non - leap year. Probability = number of favorable outcomes / total number of outcomes. So \(P(\text{Dr. Dai's BD is in August})=\frac{31}{365}\).

Step2: Find \(P(E_3)\) in sample space

In a sample space \(S = \{E_1,E_2,E_3\}\), since the sum of probabilities of all events in a sample space is 1. So \(P(E_3)=1 - P(E_1)-P(E_2)=1 - 0.45 - 0.23 = 0.32\).

Step3: Calculate empirical probability of Trump winning

In the survey results \([T,B,B,T,B,T,B,B,B,T,T,B,T,B,T,B]\), there are 16 results and the number of \(T\) (Trump) is 7. Empirical probability \(P(\text{Trump will be President})=\frac{7}{16}=0.4375 = 43.75\%\).

Step4: Find \(x\) using set - theory

We know that \(n(S)=68\), \(n(M) = 21\), \(n(E)=33\), and \(n((M\cup E)^c)=26\). First, \(n(M\cup E)=n(S)-n((M\cup E)^c)=68 - 26 = 42\). By the formula \(n(M\cup E)=n(M)+n(E)-n(M\cap E)\), we have \(42=21 + 33-x\), so \(x = 12\). Then \(P(M\cap E)=\frac{x}{n(S)}=\frac{12}{68}\approx0.1765 = 17.65\%\).

Step5: Find \(P((E\cap M)^c)\)

We know that \(P(E\cap M)=\frac{12}{68}\), and by the complement law \(P((E\cap M)^c)=1 - P(E\cap M)=1-\frac{12}{68}=\frac{68 - 12}{68}=\frac{56}{68}=\frac{14}{17}\).

Step6: Find \(P(M)\)

\(P(M)=\frac{n(M)}{n(S)}=\frac{21}{68}\approx0.3088\).

Step7: Find \(P(M\cup E)\)

We already found \(n(M\cup E) = 42\), so \(P(M\cup E)=\frac{n(M\cup E)}{n(S)}=\frac{42}{68}\).

Step8: Find \(P(M|E)\)

By the formula for conditional probability \(P(M|E)=\frac{P(M\cap E)}{P(E)}\), and \(P(M\cap E)=\frac{12}{68}\), \(P(E)=\frac{33}{68}\), so \(P(M|E)=\frac{n(M\cap E)}{n(E)}=\frac{12}{33}=\frac{4}{11}\).

Step9: Find \(P(E|M)\)

By the formula for conditional probability \(P(E|M)=\frac{P(M\cap E)}{P(M)}\), \(P(M\cap E)=\frac{12}{68}\), \(P(M)=\frac{21}{68}\), so \(P(E|M)=\frac{n(M\cap E)}{n(M)}=\frac{12}{21}=\frac{4}{7}\).

Answer:

1. D. 31/365
2. C. 0.32
3. B. 43.75%
4. C. 17.65%
5. A. 14/17
6. D. 0.3088
7. D. 42/68
8. D. 4/11
9. B. 4/7