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Independent events

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visibility 56update 6 months agobookmarkshare

🎯 In this topic you will

Learn about independent events
Use probabilities to determine whether two events are independent

🧠 Key Words

independent events

Show Definitions

independent events: Two events in probability are independent if the outcome of one does not affect the probability of the other.

🎲 Independent and Dependent Events

You flip a coin and then you roll a dice. Here are two events.

A: a head on the coin
B: a 4 on the dice

If A happens, the coin lands on a head. Then the probability of 4, P(4) = $\tfrac{1}{6}$. If A does not happen, the coin lands on a tail. Then the probability of 4, P(4), is still $\tfrac{1}{6}$. Whether A happens or not does not affect the probability of B. You say that A and B are independent events.

Now suppose you have 10 balls, numbered from 1 to 10, in a bag. You take out one ball at random. Here are two events.

C: the number is odd
D: the number is less than 4

Suppose C happens. The number is 1, 3, 5, 7 or 9. Two of these numbers are less than 4, and so P(D) = $\tfrac{2}{5}$. Now suppose C does not happen. The number is 2, 4, 6, 8 or 10. Only one of these numbers is less than 4, so now P(D) = $\tfrac{1}{5}$. The probabilities are not the same and so C and D are not independent events. Whether C happens or not does affect the probability of D.

📘 Worked example

You roll a fair dice. Here are three events.

X: the number is even
Y: the number is more than 2
Z: the number is a prime number

a. Show that X and Y are independent events.

b. Show that X and Z are not independent events.

Answer:

a. Suppose X happens. The number is 2, 4 or 6.
Two of these three numbers are more than 2, and so P(Y) = $\tfrac{2}{3}$.
Suppose X does not happen. Then the number is 1, 3 or 5.
Two of these three numbers are more than 2, and so P(Y) = $\tfrac{2}{3}$.
P(Y) has not changed, and so X and Y are independent.

b. Suppose X happens. The number is 2, 4 or 6.
Only one of these numbers is a prime number, and so P(Z) = $\tfrac{1}{3}$.
Suppose X does not happen. Then the number is 1, 3 or 5.
Two of these numbers (3 and 5) are prime numbers, and so P(Z) = $\tfrac{2}{3}$.
P(Z) is not the same in both cases, and so the events X and Z are not independent.

To test independence, compare probabilities when the first event happens and when it does not. If the probability of the second event remains the same, the two events are independent. If it changes, they are not independent.

❓ EXERCISES

1. A coin is flipped twice. Here are two events.
F: the first flip is a head S: the second flip is a head
Explain why F and S are independent events.

👀 Show answer

The outcome of the first flip does not affect the second flip. Each flip has probability $\tfrac{1}{2}$ of being a head. Therefore, F and S are independent.

2. A fair dice is rolled. Here are two events.
A: the number is 2, 3 or 4 B: the number is 1 or 2
Show that A and B are independent events.

👀 Show answer

P(A) = $\tfrac{3}{6} = \tfrac{1}{2}$, P(B) = $\tfrac{2}{6} = \tfrac{1}{3}$. P(A ∩ B) = {2} = $\tfrac{1}{6}$. Since $P(A)P(B) = \tfrac{1}{2}\times\tfrac{1}{3} = \tfrac{1}{6}$, A and B are independent.

3. A coin is flipped three times. Here are two events.
X: the first two flips are tails Y: all three flips are tails
Are X and Y independent events? Give a reason for your answer.

👀 Show answer

P(X) = $\tfrac{1}{4}$, P(Y) = $\tfrac{1}{8}$. P(X ∩ Y) = probability of all 3 tails = $\tfrac{1}{8}$. But $P(X)P(Y) = \tfrac{1}{4}\times\tfrac{1}{8} = \tfrac{1}{32}$, which is not equal to $\tfrac{1}{8}$. So X and Y are not independent.

4. A fair coin is flipped ten times. Here are two events.
A: the first nine flips are heads B: the tenth flip is a head
Are A and B independent? Give a reason for your answer.

👀 Show answer

The tenth flip is still independent of the first nine. P(B) = $\tfrac{1}{2}$ regardless of A. So A and B are independent.

5. Here are two events.
A: there is fog at the airport B: the flight to Dubai leaves on time
Explain why these events are not independent.

👀 Show answer

Fog at the airport makes a delay more likely. The probability of B depends on A. Therefore, A and B are not independent.

❓ EXERCISES

6. There are ten cards in a pack.
Six cards have the numbers 1, 2, 3, 4, 5, 6 in red.
Four cards have the numbers 1, 2, 3, 4 in black.

a. Here are two events.
R: the number is red E: the number is even
Are these independent events? Give a reason for your answer.

👀 Show answer

P(R) = $\tfrac{6}{10} = \tfrac{3}{5}$, P(E) = $\tfrac{5}{10} = \tfrac{1}{2}$. P(R ∩ E) = {2, 4, 6 red} = $\tfrac{3}{10}$. Since $P(R)P(E) = \tfrac{3}{5}\times\tfrac{1}{2} = \tfrac{3}{10}$, the events are independent.

b. Here are two events.
B: the number is black T: the number is 2
Are these independent events? Give a reason for your answer.

👀 Show answer

P(B) = $\tfrac{4}{10} = \tfrac{2}{5}$, P(T) = $\tfrac{2}{10} = \tfrac{1}{5}$. P(B ∩ T) = only the black 2 = $\tfrac{1}{10}$. Since $P(B)P(T) = \tfrac{2}{5}\times\tfrac{1}{5} = \tfrac{2}{25}$, which is not equal to $\tfrac{1}{10}$, the events are not independent.

7. There are ten balls in a bag. Three balls are black and seven balls are white.

a. One ball is chosen at random and then replaced. Then a second ball is chosen at random.
Here are two events.
F: the first ball is black S: the second ball is black
Are F and S independent? Give a reason for your answer.

👀 Show answer

With replacement, the second draw is unaffected by the first. P(F) = $\tfrac{3}{10}$, P(S) = $\tfrac{3}{10}$. The probability remains the same regardless of the first draw, so F and S are independent.

b. The situation is the same as in part a, but this time the first ball is not replaced.
Are F and S independent in this case? Give a reason for your answer.

👀 Show answer

Without replacement, the second draw depends on the first. If F happens, fewer black balls remain. The probabilities change, so F and S are not independent.

8. Arun and Sofia attend the same school. Here are two events.
A: Arun is late for school S: Sofia is late for school

a. Describe how A and S could be independent events.

👀 Show answer

If Arun and Sofia travel separately and their lateness depends only on their own choices, then A and S are independent.

b. Describe how A and S could be events that are not independent.

👀 Show answer

If Arun and Sofia travel together (e.g., same bus), then if one is late the other is also likely to be late. So A and S are not independent.

9. Here are five cards.

A card is chosen at random. Here are two events.
X: the letter is in the word CARD Y: the letter is in the word CODE
Are these events independent? Give a reason for your answer, using probabilities.

👀 Show answer

P(X) = $\tfrac{4}{5}$ (A, B, C, D). P(Y) = $\tfrac{3}{5}$ (C, D, E). P(X ∩ Y) = {C, D} = $\tfrac{2}{5}$. P(X)P(Y) = $\tfrac{4}{5}\times\tfrac{3}{5} = \tfrac{12}{25}$. Since $\tfrac{12}{25}\neq\tfrac{2}{5}$, the events are not independent.

📘 What we've learned

We understood that two events are independent if the occurrence of one does not affect the probability of the other.
We explored real-life and mathematical examples (coin flips, dice rolls, drawing balls/cards) to test independence.
The formal rule is: events $A$ and $B$ are independent if $P(A \cap B) = P(A) \times P(B)$.
If $P(A \cap B) \neq P(A) \times P(B)$, then the events are not independent.
We practiced calculating probabilities with and without replacement, and explained why this affects independence.
We learned how to reason about independence in everyday contexts (e.g., lateness, weather, flight delays).

Independent events

🎯 In this topic you will

🧠 Key Words

🎲 Independent and Dependent Events

❓ EXERCISES

❓ EXERCISES

📘 What we've learned

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