DAT - Dental Admission Quantitative Reasoning: Probability Questions and Answers

Question 1

A dental clinic uses two independent sterilization machines, A and B. Machine A has a 95% success rate, and Machine B has a 90% success rate. If a set of instruments is processed through both machines, what is the probability that at least one machine successfully sterilizes the set?

Accepted Answer

0.995

Answer

To find the probability of at least one success, it's easiest to first calculate the probability of the complementary event, which is that both machines fail. Then, subtract this from 1.
Probability of Machine A failing = 1 - 0.95 = 0.05.
Probability of Machine B failing = 1 - 0.90 = 0.10.
Since the events are independent, the probability of both failing is the product of their individual probabilities: 0.05 * 0.10 = 0.005.
The probability of at least one machine succeeding is 1 - P(both fail) = 1 - 0.005 = 0.995.

Question 2

A dental study group consists of 5 first-year students and 4 second-year students. If a committee of 4 students is selected at random, what is the probability that the committee will consist of exactly 2 first-year students and 2 second-year students?

Accepted Answer

10/21

Answer

First, calculate the total number of ways to choose a committee of 4 from the 9 students using combinations: C(9,4) = 9! / (4! * 5!) = (9*8*7*6)/(4*3*2*1) = 126 total possible committees.
Next, calculate the number of ways to choose 2 first-year students from 5: C(5,2) = 5! / (2! * 3!) = 10.
Then, calculate the number of ways to choose 2 second-year students from 4: C(4,2) = 4! / (2! * 2!) = 6.
The number of ways to form the specific committee is the product of these two results: 10 * 6 = 60 favorable outcomes.
The probability is the ratio of favorable outcomes to total outcomes: 60/126, which simplifies to 10/21.

Question 3

In a standard 52-card deck, what is the probability of drawing a single card that is a King or a Spade?

Accepted Answer

4/13

Answer

This problem involves non-mutually exclusive events because a card can be both a King and a Spade (the King of Spades). The formula is P(A or B) = P(A) + P(B) - P(A and B).
P(King) = 4/52.
P(Spade) = 13/52.
P(King and Spade) = 1/52.
Therefore, P(King or Spade) = (4/52) + (13/52) - (1/52) = 16/52. This fraction simplifies to 4/13.

Question 4

A bag contains 6 red marbles, 4 blue marbles, and 5 yellow marbles. If one marble is drawn at random, which of the following events has the highest probability of occurring?

Accepted Answer

Drawing a marble that is not yellow

Answer

First, find the total number of marbles: 6 + 4 + 5 = 15.
Now, calculate the probability of each event:
A) P(Red) = 6/15 = 0.40
B) P(Blue) = 4/15 ≈ 0.27
C) P(Not Red) = P(Blue or Yellow) = (4+5)/15 = 9/15 = 0.60
D) P(Not Yellow) = P(Red or Blue) = (6+4)/15 = 10/15 ≈ 0.67
Comparing the probabilities, drawing a marble that is not yellow (0.67) has the highest probability.

Question 5

A fair six-sided die is rolled three times. What is the probability of rolling a '4' on the first roll, an even number on the second roll, and a number greater than 2 on the third roll?

Accepted Answer

1/18

Answer

These are three independent events, so their combined probability is the product of their individual probabilities.
1. Probability of rolling a '4' = 1/6.
2. Probability of rolling an even number ({2, 4, 6}) = 3/6 = 1/2.
3. Probability of rolling a number greater than 2 ({3, 4, 5, 6}) = 4/6 = 2/3.
Multiply the probabilities together: (1/6) * (1/2) * (2/3) = 2/36 = 1/18.

DAT Practice Test

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DAT - Dental Admission Quantitative Reasoning: Probability Questions and Answers