Probability questions on Digital SAT Math split into two main types: straightforward probability (favorable outcomes divided by total outcomes) and conditional probability, which asks for the chance of one event given that another has already happened. Conditional probability problems almost always come with a two-way data table, and the entire skill comes down to correctly identifying which row, column, or sub-total to use as the new "total" once the condition restricts the sample space.
The 13 problems below cover this full range, including basic probability from single categories, missing-value problems where you first have to find an unknown group size, and several conditional probability problems built on two-way tables. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. For any "given that" conditional probability problem, the fastest and most reliable approach is to physically cross out the rows or columns that the condition eliminates, then treat what remains as an entirely new, smaller table. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
A certain book has 250 pages, and 21 of these pages have an illustration. If one of the book's pages is selected at random, what is the probability of selecting a page with an illustration? (Express your answer as a decimal or fraction, not as a percent.)
Divide the number of favorable outcomes (illustrated pages) by the total number of outcomes (total pages): 21/250.
21/250 = 0.084.
At a convention center, there are a total of 325 visitors. Each visitor is located in either room A, room B, or room C. If one of these visitors is selected at random, the probability of selecting a visitor who is located in room A is 0.48, and the probability of selecting a visitor who is located in room B is 0.16. How many visitors are located in room C?
Find the number of visitors in each named room: room A has 0.48 × 325 = 156 visitors, and room B has 0.16 × 325 = 52 visitors.
Subtract both from the total to find room C: 325 − 156 − 52 = 117. The correct answer is D.
There are 20 marbles in the bag: 8 red marbles, 10 green marbles, and 2 blue marbles. If one of these marbles is selected at random, what is the probability of selecting a red marble?
Divide the number of red marbles by the total number of marbles: 8/20.
The correct answer is B.
A choir of 48 singers has 12 members who sing bass. If one singer from the choir is selected at random, what is the probability of selecting a singer who sings bass?
Divide the number of bass singers by the total number of choir members: 12/48.
The correct answer is B.
The table shows the distribution of people in a certain city by age group. If a person in this city is selected at random, which of the following is closest to the probability of selecting a person who is greater than 65 years old, given that the person is at least 18 years old?
| Age group | Proportion |
|---|---|
| Less than 18 years old | 27% |
| 18–40 years old | 24% |
| 41–65 years old | 28% |
| Greater than 65 years old | 21% |
"At least 18 years old" excludes the "less than 18" group, leaving 24% + 28% + 21% = 73% of the population as the new total.
Within that group, the "greater than 65" proportion is 21%. The conditional probability is 21/73 ≈ 0.29. The correct answer is B.
At a convention center, there are a total of 375 visitors. Each visitor is located in either room A, room B, or room C. If one of these visitors is selected at random, the probability of selecting a visitor who is located in room A is 0.48, and the probability of selecting a visitor who is located in room B is 0.32. How many visitors are located in room C?
Find the number of visitors in each named room: room A has 0.48 × 375 = 180 visitors, and room B has 0.32 × 375 = 120 visitors.
Subtract both from the total to find room C: 375 − 180 − 120 = 75. The correct answer is D.
A sleep study consisted of 59 participants, of which 53 participants each had an average of more than 120 minutes of rapid eye movement (REM) sleep per night. If a participant from this study is selected at random, what is the probability of selecting a participant that had an average of more than 120 minutes of REM sleep per night?
Divide the number of qualifying participants by the total number of participants: 53/59.
The correct answer is C.
The table summarizes the distribution of age and assigned group for participants in a study. One of these participants will be selected at random. What is the probability of selecting a participant from group A, given that the participant is at least 10 years of age? (Express your answer as a decimal or fraction, not as a percent.)
| 0-9 years | 10-19 years | 20+ years | Total | |
|---|---|---|---|---|
| Group A | 14 | 19 | 27 | 60 |
| Group B | 27 | 7 | 26 | 60 |
| Group C | 19 | 34 | 7 | 60 |
| Total | 60 | 60 | 60 | 180 |
"At least 10 years old" combines the 10-19 and 20+ columns: 60 + 60 = 120 participants total in that restricted group.
Within that group, group A contributes 19 (from 10-19) + 27 (from 20+) = 46 participants. The conditional probability is 46/120 = 23/60.
A bin contains a mixture of T-shirts for two sports teams. The table shows the number of T-shirts in the bin, classified by size and sports team. One T-shirt from the bin will be selected at random. What is the probability of selecting a T-shirt that is medium, given that the T-shirt is a Flames T-shirt? (Express your answer as a decimal or fraction, not as a percent.)
| Flames | Panthers | Total | |
|---|---|---|---|
| Small | 6 | 9 | 15 |
| Medium | 24 | 33 | 57 |
| Large | 18 | 24 | 42 |
| Total | 48 | 66 | 114 |
"Given that the T-shirt is a Flames T-shirt" restricts the sample space to just the Flames column, which totals 48 T-shirts.
Within that column, 24 are medium. The conditional probability is 24/48 = 0.5.
The table summarizes the distribution of age and assigned group for 105 participants in a study. One of these participants will be selected at random. What is the probability of selecting a participant from group A, given that the participant is at least 10 years of age?
| 0-9 years | 10-19 years | 20+ years | Total | |
|---|---|---|---|---|
| Group A | 5 | 19 | 11 | 35 |
| Group B | 4 | 8 | 23 | 35 |
| Group C | 26 | 8 | 1 | 35 |
| Total | 35 | 35 | 35 | 105 |
"At least 10 years old" combines the 10-19 and 20+ columns: 35 + 35 = 70 participants total in that restricted group.
Within that group, group A contributes 19 (from 10-19) + 11 (from 20+) = 30 participants. The conditional probability is 30/70 = 3/7. The correct answer is B.
The table summarizes members of a local organization by age and whether they live north or south of Center St. If a member of the organization is selected at random, what is the probability that the selected member is at least 45 years old?
| Live north of Center St. | Live south of Center St. | Total | |
|---|---|---|---|
| Less than 45 years old | 13 | 12 | 25 |
| At least 45 years old | 22 | 88 | 110 |
| Total | 35 | 100 | 135 |
This is a straightforward (non-conditional) probability, using the full total of 135 members. The "at least 45 years old" row totals 110 members.
The probability is 110/135. The correct answer is D.
A researcher calculated the mitochondrial genome sizes g, in kilobase pairs (kbp), and the number of mitochondrial genes for 34 different species. The table summarizes these data. One of these species is selected at random. What is the probability of selecting a species with a mitochondrial genome size greater than or equal to 18 kbp, given that it has 37 or more mitochondrial genes? (Express your answer as a decimal or fraction, not as a percent.)
| Number of mitochondrial genes | g < 16 | 16 ≤ g < 18 | 18 ≤ g < 20 | g ≥ 20 | Total |
|---|---|---|---|---|---|
| 36 or less | 3 | 0 | 0 | 0 | 3 |
| 37 | 1 | 19 | 0 | 0 | 20 |
| 38 or more | 2 | 0 | 4 | 0 | 6 |
| Total | 6 | 19 | 4 | 0 | 29 |
"37 or more mitochondrial genes" combines the "37" and "38 or more" rows: 20 + 6 = 26 species total in that restricted group.
Within that group, genome size ≥ 18 kbp means the "18 ≤ g < 20" and "g ≥ 20" columns: for the "37" row that's 0 + 0 = 0, and for the "38 or more" row that's 4 + 0 = 4, giving 4 species total. The conditional probability is 4/26 = 2/13.
For 100 neurons, the table summarizes the distribution of classification and cell body diameter. One of these neurons will be selected at random. What is the probability of selecting a neuron with a cell body diameter that is less than or equal to 30 micrometers, given that it is not classified as a motor neuron?
| Classification | Cell body diameter (micrometers) | ||
|---|---|---|---|
| Less than 20 | 20 to 30 | Greater than 30 | |
| Sensory neuron | 13 | 7 | 3 |
| Motor neuron | 0 | 15 | 18 |
| Interneuron | 10 | 34 | 0 |
"Not classified as a motor neuron" combines the sensory neuron row (total 13+7+3=23) and interneuron row (total 10+34+0=44), giving 23 + 44 = 67 neurons total in that restricted group.
Within that group, diameter ≤ 30 means the "less than 20" and "20 to 30" columns: for sensory that's 13+7=20, and for interneuron that's 10+34=44, giving 64 neurons total. The conditional probability is 64/67.
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