Practice Probabilistic Thinking in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Probabilistic thinking is the habit of reasoning about uncertain outcomes in terms of likelihood, expected value, and distributions rather than certainties.

Instead of 'Will X happen?' ask 'How likely is X?' and plan for multiple outcomes.

Showing a random 20 of 50 problems.

Example 1

easy
You roll a die and it shows 1, 2, 3, 4, 5 five times in a row (never 6). What is the probability of rolling a 6 on the next roll? Explain the correct probabilistic reasoning.

Example 2

medium
Expected value: a game pays $10 with probability 0.20.2 and $0 otherwise. What is the expected payout?

Example 3

medium
A weather app says 80%80\% chance of rain for 55 days. Should you assume it will rain all 55 days?

Example 4

hard
Bayes' base-rate problem: 11 in 10001000 people have a disease. A test is 99%99\% accurate (both ways). You test positive. Estimate P(disease∣positive)P(\text{disease} \mid \text{positive}).

Example 5

medium
Drawing 22 cards from a standard deck without replacement, what is the probability both are aces?

Example 6

medium
Independent events: P(A)=0.5P(A) = 0.5, P(B)=0.4P(B) = 0.4. Find P(A and B)P(A \text{ and } B) and P(A or B)P(A \text{ or } B).

Example 7

easy
One friend says a treatment 'worked for me.' Population data shows it works 20%20\% of the time. Which should guide your expectation?

Example 8

medium
The gambler's fallacy: after 44 reds at roulette, a player bets big on black 'because it's due.' What is the probability of black on the next spin (European wheel, 1818 black of 3737)?

Example 9

medium
You face two bets, each with 50%50\% chance of winning. Bet A: win $1 or lose $1. Bet B: win $1000 or lose your savings. Are these the "same" because both are 50%50\% chances?

Example 10

easy
A fair die is rolled. What is the probability of rolling a 44?

Example 11

easy
Two events: '50%50\% chance to win $1' vs '50%50\% chance to lose your house.' Are these 'the same' because both are coin flips?

Example 12

easy
A standard die is rolled. Find P(roll greater than 4)P(\text{roll greater than } 4).

Example 13

medium
A coin lands heads 77 times in a row. What is P(next flip is heads)P(\text{next flip is heads}) for a fair coin, and what cognitive bias does the wrong answer reflect?

Example 14

medium
A bag contains 44 red, 55 blue, and 11 green marble. Find P(red or green)P(\text{red or green}).

Example 15

hard
A class has 3030 students. Roughly, what is the probability that two share a birthday (ignoring leap years)?

Example 16

medium
A driver claims "I've never had an accident in 2020 years, so I'm safe." What probabilistic flaw does this reasoning have?

Example 17

easy
Why is thinking '70%70\% chance' better than 'it will probably happen' for planning?

Example 18

easy
A spinner has 44 equal sectors numbered 11 through 44. Find P(landing on an even number)P(\text{landing on an even number}).

Example 19

hard
The Monty Hall Problem: 3 doors, prize behind 1. You pick door 1. Host opens door 3 (no prize). Should you switch to door 2? Calculate probabilities for staying vs. switching.

Example 20

easy
A bag has 77 red and 33 blue marbles. What is the probability that a random marble is blue?
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