D. Data · Data Literacy and Probability

Independent and Dependent Events

D2.1About 45 minutes

Learning goals

I can explain the difference between independent and dependent events.
I can explain how the probability of the second event changes, or doesn't change, after the first event.
I can give my own examples of independent and dependent events.

Words to know

probability
A number from 0 to 1 (or 0% to 100%) that tells how likely an event is.

independent events
Two events where the result of the first does not change the probability of the second.

dependent events
Two events where the result of the first does change the probability of the second.

with replacement
Putting an item back before choosing again. The choices stay the same.

without replacement
Keeping an item out after choosing it. There is one fewer item for the next choice.

Remember from Grade 6

You found theoretical probabilities as fractions, decimals and percents, and you found the probability of two independent events, like flipping a coin and spinning a spinner. Today you'll meet dependent events, where the first event changes what can happen next.

Does the first event matter?

Ava has a bag of marbles. She picks one without looking, then picks again. Does her first pick change the chances for her second pick?

It depends on one thing: does she put the first marble back?

Three green marbles and two blue marbles in front of a small cloth drawstring bag on a wooden table. — A bag of 3 green and 2 blue marbles, like Ava's. Putting a marble back or keeping it out changes the chances for the next pick. AI-generated image

Three green marbles labelled G and two blue marbles labelled B in a bag. 5 marbles in all.

P(blue) on the first pick = 2/5, because 2 of the 5 marbles are blue.

If Ava puts the marble back (with replacement), the bag is the same as before. P(blue) on the second pick is still 2/5. The events are independent.
If Ava keeps the marble out (without replacement), the bag has changed. P(blue) on the second pick depends on what she picked first. The events are dependent.

Left: the blue marble goes back, so the bag still has 3 G and 2 B, and P(blue) stays 2/5. Right: the blue marble stays out, so the bag has 3 G and 1 B, and P(blue) becomes 1/4.

Worked example

Example 1: With replacement (independent)

Ava picks a marble, notes the colour, puts it back, shakes the bag and picks again. Her first marble was blue. What is P(blue) on the second pick?

The marble went back, so the bag again has 3 green and 2 blue: 5 marbles.
P(blue) = 2/5 = 0.4 = 40%.
It would also be 2/5 if her first marble had been green.

2/5 (40%). The first pick didn't change anything. The events are independent.

Worked example

Example 2: Without replacement (dependent)

Ava picks a marble and keeps it out. Then she picks again. Find P(blue on the second pick) if (a) the first was blue, (b) the first was green.

(a) First was blue: the bag has 3 green and 1 blue, 4 marbles. P(blue) = 1/4 = 25%.
(b) First was green: the bag has 2 green and 2 blue, 4 marbles. P(blue) = 2/4 = 1/2 = 50%.
The probability for the second pick is different depending on the first pick.

(a) 1/4 (25%), (b) 1/2 (50%). The events are dependent.

Independent events	Dependent events
The first result does not change the probabilities for the second.	The first result does change the probabilities for the second.
Flip a coin, then roll a die.	Pick a card, keep it, pick another card.
Pick a marble, put it back, pick again.	Pick a marble, keep it out, pick again.
Spin a spinner twice.	Draw two names from a hat for two class jobs (the first name is not put back).
Roll a die twice.	Pull two socks, one after the other, from a drawer in the dark.

Some examples of each

Worked example

Example 3: Sort the situations

Independent or dependent?

(a) Rolling a 6 on a die, then flipping heads on a coin. (b) Choosing a captain and then an assistant captain from 12 hockey players. (c) Winning a door prize at a school event, then winning a second prize when your ticket is put back in the draw.

(a) The die can't affect the coin. Independent.
(b) Once the captain is chosen, only 11 players are left for assistant captain. The first choice changes the second. Dependent.
(c) The ticket goes back in, so the draw is the same as before. Independent.

(a) independent, (b) dependent, (c) independent.

The coin has no memory

Flip a fair coin and get heads 5 times in a row. Is tails "due" next? No! Each flip is independent, so P(tails) is still 1/2. Many people believe the opposite. Thinking it through with the idea of independence helps you avoid this trap.

Just for fun

Why should a sock drawer win a probability award?

Every sock you pull out in the dark changes the odds for the next one: dependent events in action!

Hands-on activity

Cards in and out

You need: a deck of playing cards and a grown-up or a sibling.

Take out 3 hearts and 2 spades. Shuffle these 5 cards and lay them face down.
With replacement: pick a card, record it, put it back, shuffle, pick again. What is P(heart) on the second pick? (3/5 every time.)
Without replacement: pick a card, keep it face up beside you, then pick a second card. Work out P(heart) on the second pick: if the first was a heart, 2 hearts are left out of 4 cards (2/4); if the first was a spade, 3 hearts are left out of 4 cards (3/4).
Make up your own example of independent events and dependent events from everyday life. Explain each one to your partner.

Math talk: "Why does keeping the first card change the chances?"

Your turn

Independent or dependent? Spin a spinner, then spin it again.

Independent or dependent? Pick 2 students from a class of 25 to hand out books; the same student can't be picked twice.

A bag has 4 red and 1 yellow tile. You pick a tile and put it back. What is P(yellow) on the next pick?

Same bag (4 red, 1 yellow). You pick the yellow tile and keep it out. What is P(yellow) on the next pick?

Same bag. You pick a red tile and keep it out. What is P(yellow) on the next pick?

A coin landed on heads 6 times in a row. What is P(heads) on the next flip?

Give your own example of dependent events.

More practice: warm-up

Rolling a die, then flipping a coin: independent or dependent?

Drawing two cards from a deck without putting the first back: independent or dependent?

A bag has 2 green and 2 blue marbles. P(green)?

What does "with replacement" mean?

Does a die "remember" what it rolled last time?

More practice: on your own

A box has 5 vanilla and 3 chocolate granola bars. Eli takes one and eats it; it was chocolate. What is P(chocolate) for the next bar? Before he took one, what was it?

A spinner has 4 equal sections: 1, 2, 3, 4. Kai spins 3 twice in a row. What is P(3) on the third spin?

In a class draw, 20 names go in a hat. Two prizes are drawn without putting names back. Leah's name wasn't drawn first. What is her chance of being drawn second?

Explain why picking socks from a drawer in the dark, one after another, gives dependent events.

A bag has 3 green and 2 blue marbles. Ava picks blue and keeps it out. Then she picks blue again and keeps it out. What is P(blue) on her third pick?

Challenge

Is this independent or dependent? "A weather forecast says there is a 60% chance of rain tomorrow. Whether it rains today doesn't matter." Explain why weather is tricky.

A bag has some marbles. P(green) on the first pick is 1/2. Ava picks a green marble and keeps it out. Now P(green) is 2/5. How many marbles were in the bag at first, and how many were green?

Explain how you could change the dependent event "pick two cards from 5 cards" into independent events.

Check your understanding

Quiz

Which pair of events is independent?

What makes two events dependent?

A bag has 3 green and 2 blue marbles. You pick a green marble and keep it out. What is P(green) on the next pick?

Same bag (3 green, 2 blue). You pick a green marble and put it back. What is P(green) on the next pick?

A fair coin lands on tails 4 times in a row. What is P(heads) on the next flip?

Choosing a class president and then a vice-president from the same 20 students is:

For parents and teachers

Big idea: D2.1 asks Grade 7 students to describe the difference between independent and dependent events and explain how their probabilities differ, with examples. In Grade 6 they worked only with independent events. The simplest test: "Does the first result change the chances for the second?" Putting an item back (with replacement) usually keeps events independent; keeping it out (without replacement) makes them dependent.

Common difficulties:

The "due" belief (after many heads, tails is due). Coins, dice and spinners have no memory.
Forgetting to reduce the total after an item is kept out.
Calling events dependent just because they happen one after the other.

How to help: Use a small set of cards, socks or wrapped candies. Ask your child to predict the chances before and after one item is taken. Look for real examples: drawing names for gift exchanges, picking teams, choosing toppings.