Independent and dependent events
Classify events as independent or dependent and calculate probabilities using the multiplication rule, adjusting for dependence where required.
Worked examples
Classifying events as independent or dependent
Straightforward
Problem
A bag contains 6 blue counters and 4 red counters. Consider the following two situations.
**Situation A:** A counter is drawn, its colour is noted, and it is then replaced before a second counter is drawn.
**Situation B:** A counter is drawn and set aside. A second counter is then drawn from the remaining counters.
Classify each situation as involving independent or dependent events, and explain your reasoning.
**Situation A:** A counter is drawn, its colour is noted, and it is then replaced before a second counter is drawn.
**Situation B:** A counter is drawn and set aside. A second counter is then drawn from the remaining counters.
Classify each situation as involving independent or dependent events, and explain your reasoning.
1
Consider Situation A: drawing with replacement.
In Situation A, the counter is returned to the bag after the first draw. This means the bag always contains 6 blue and 4 red counters — a total of 10 counters — at the start of each draw. The outcome of the first draw has no effect on the contents of the bag for the second draw.
2
Classify Situation A.
Because the probabilities on the second draw are unchanged by the first draw, the two events in Situation A are **independent**.
3
Consider Situation B: drawing without replacement.
In Situation B, the first counter is not returned. If the first counter drawn was blue, there are now 5 blue and 4 red counters left (9 counters in total). If the first counter was red, there are 6 blue and 3 red counters left. Either way, the composition of the bag has changed, and the probabilities for the second draw are different depending on what was drawn first.
4
Classify Situation B.
Because the outcome of the first draw changes the probabilities for the second draw, the two events in Situation B are **dependent**.
Answer
Situation A (with replacement): independent events. Situation B (without replacement): dependent events.
Probability of two independent events
Moderate
Problem
A bag contains 3 green counters and 2 yellow counters. A counter is drawn, its colour is recorded, and it is then **replaced**. A second counter is drawn. Find the probability of drawing a green counter on the first draw and a yellow counter on the second draw.
1
Identify whether the events are independent or dependent.
The counter is replaced after the first draw, so the bag always contains 5 counters (3 green, 2 yellow) for each draw. The outcome of the first draw does not affect the second draw. The two events are **independent**.
2
Find the probability of a green counter on the first draw.
3
Find the probability of a yellow counter on the second draw.
Because the counter is replaced, the bag is the same for the second draw.
4
Apply the multiplication rule for independent events.
Answer
Probability of two dependent events
Challenging
Problem
A bag contains 4 red balls and 3 blue balls. Two balls are drawn **without replacement**. Find the probability that the first ball is red and the second ball is blue.
1
Identify whether the events are independent or dependent.
The first ball is not replaced, so the contents of the bag change after the first draw. The probability for the second draw depends on the outcome of the first draw. The two events are **dependent**.
2
Find the probability that the first ball is red.
There are balls in total.
3
Find the probability that the second ball is blue, given the first was red.
After one red ball is removed, the bag contains 3 red balls and 3 blue balls — a total of 6 balls.
4
Multiply the probabilities to find the combined probability.
Answer
Practise
Q1·Straightforward
A bag contains 5 red balls and 3 blue balls. One ball is drawn, its colour is recorded, and it is then **replaced** before a second ball is drawn. Are the two draws independent or dependent events?
Explanation
Because the ball is replaced after the first draw, the bag contains the same 5 red and 3 blue balls for the second draw. The outcome of the first draw has no effect on the probabilities for the second draw, so the two events are independent.
Q2·Straightforward
A card is drawn from a standard 52-card deck and **not replaced**. A second card is then drawn from the remaining 51 cards. Are the two draws independent or dependent events?
Explanation
After the first card is removed, the deck has only 51 cards and its composition has changed. The probability of any particular card on the second draw depends on what was drawn first. The two draws are dependent events.
Q3·Straightforward
A fair coin is flipped and then flipped a second time. Are the outcomes of the two flips independent or dependent events?
Explanation
A coin has no memory. Whether it lands heads or tails on the first flip has no effect on the probability of heads or tails on the second flip. The two outcomes are independent events.
Q4·Straightforward
Two students are chosen at random, one after the other, from a class of 25 students. Once a student is chosen, they cannot be chosen again. Are the two selections independent or dependent events?
Explanation
After the first student is chosen, only 24 students remain and the probabilities change. The outcome of the first selection affects the probabilities for the second selection, so the two events are dependent.
Q5·Moderate
A fair coin is flipped and a standard six-sided die is rolled at the same time. Find the probability of getting a head on the coin **and** a 3 on the die.
/
Explanation
The coin flip and die roll are independent events.
Q6·Moderate
The probability of rain on any given day in a particular city is . Assuming daily weather outcomes are independent, find the probability that it rains on both Monday and Tuesday.
Explanation
The weather on each day is independent.
The probability of rain on both days is , or .
The probability of rain on both days is , or .
Q7·Moderate
A spinner has 4 equal sections numbered 1, 2, 3 and 4. The spinner is spun twice. Find the probability of getting an even number on both spins.
/
Explanation
The two spins are independent. The even numbers on the spinner are 2 and 4.
Q8·Moderate
A bag contains 3 green counters and 2 yellow counters. A counter is drawn at random, its colour is recorded, and it is then **replaced**. A second counter is then drawn. Find the probability of drawing a green counter on both draws.
/
Explanation
Because the counter is replaced, the two draws are independent. There are 5 counters in total.
Q9·Challenging
A bag contains 4 red balls and 3 blue balls. Two balls are drawn **without replacement**. Find the probability that both balls are red.
/
Explanation
The two draws are dependent because the first ball is not replaced.
There are 7 balls in total.
Given the first ball was red, there are now 3 red balls and 3 blue balls remaining (6 balls in total).
There are 7 balls in total.
Given the first ball was red, there are now 3 red balls and 3 blue balls remaining (6 balls in total).
Q10·Challenging
A box contains 5 white marbles and 4 black marbles. Two marbles are drawn **without replacement**. Find the probability that both marbles are black.
/
Explanation
The two draws are dependent because marbles are not replaced.
There are 9 marbles in total.
Given the first marble was black, there are now 3 black marbles and 5 white marbles remaining (8 marbles in total).
There are 9 marbles in total.
Given the first marble was black, there are now 3 black marbles and 5 white marbles remaining (8 marbles in total).
Q11·Challenging
A class has 10 boys and 8 girls. Two students are chosen at random, one at a time, **without replacement**. Find the probability that both students chosen are boys.
/
Explanation
The two selections are dependent because students are not replaced.
There are students in total.
Given the first student was a boy, there are now 9 boys and 8 girls remaining (17 students in total).
There are students in total.
Given the first student was a boy, there are now 9 boys and 8 girls remaining (17 students in total).
Q12·Challenging
Two cards are dealt one at a time from a standard shuffled deck of 52 cards, **without replacement**. Find the probability that both cards are aces.
/
Explanation
The two draws are dependent because cards are not replaced.
There are 4 aces in a deck of 52 cards.
Given the first card was an ace, there are now 3 aces remaining in a deck of 51 cards.
There are 4 aces in a deck of 52 cards.
Given the first card was an ace, there are now 3 aces remaining in a deck of 51 cards.