Steply

Sample spaces

Probability · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. List all outcomes
  2. Count the ones we want
  3. Write the fraction
  4. Check

The steps for Flip 2 coins, P(two heads). Every solution names its steps, so you always know what comes next.

Worked examples

Flip 2 coins, P(two heads) Easy

The question

Question: 2 coins are flipped.

What is the probability of getting 2 heads?

The probability is a fraction: the outcomes we want, over all the outcomes.

P(2 heads) = we wantall

P stands for probability.

P = probability

1 List all outcomes

Each coin lands heads, H, or tails, T.

Here is every outcome.

Each coin has 2 ways.

2 times 2 is 4 outcomes, all just as likely.

all = 2 × 2 = 4

This list of all the outcomes is called the sample space.

H = heads, T = tails
sample space = all the outcomes

2 Count the ones we want

Outcomes with 2 heads: just one.

That is 1 outcome we want.

we want = 1

3 Write the fraction

1 we want, out of 4.

So the probability is 1/4.

P(2 heads) = 14

Careful!

0, 1 or 2 heads are not equally likely.

HT and TH are two different outcomes.

Careful: not 13 ✗

4 Check

Let's check with the others.

4 minus 1 is 3.

1/4 plus 3/4 is 4/4: that is 1, all of them.

So the answer is right!

P(2 heads) + P(not 2 heads) = 14 + 34 = 44 = 1

Let's look back.

First: list all outcomes.

Then: count the ones we want.

Then: write the fraction.

Last: check.

Answer
Careful: not 13 ✗

▶ Watch it solved

Flip 3 coins, P(exactly 2 heads) Medium

The question

Question: 3 coins are flipped.

What is the probability of getting exactly 2 heads?

The probability is a fraction: the outcomes we want, over all the outcomes.

P(exactly 2 heads) = we wantall

P stands for probability.

P = probability

1 List all outcomes

Each coin lands heads, H, or tails, T.

Here is every outcome.

Each coin has 2 ways.

2 times 2 times 2 is 8 outcomes, all just as likely.

all = 2 × 2 × 2 = 8

This list of all the outcomes is called the sample space.

H = heads, T = tails
sample space = all the outcomes

2 Count the ones we want

Outcomes with exactly 2 heads: HHT, HTH and THH.

That is 3 outcomes we want.

we want = 3

3 Write the fraction

3 we want, out of 8.

So the probability is 3/8.

P(exactly 2 heads) = 38

Careful!

0, 1, 2 or 3 heads are not equally likely.

HT and TH are two different outcomes.

Careful: not 14 ✗

4 Check

Let's check with the others.

8 minus 3 is 5.

3/8 plus 5/8 is 8/8: that is 1, all of them.

So the answer is right!

P(exactly 2 heads) + P(not exactly 2 heads) = 38 + 58 = 88 = 1

Let's look back.

First: list all outcomes.

Then: count the ones we want.

Then: write the fraction.

Last: check.

Answer
Careful: not 14 ✗

▶ Watch it solved

Flip 3 coins, P(at least 1 head) Harder

The question

Question: 3 coins are flipped.

What is the probability of getting at least 1 head?

At least 1 head means every outcome except no heads.

So we find the probability of no heads first.

Then we take it from 1.

P(at least 1 head) = 1 − P(no heads)

P stands for probability.

All the outcomes together make 1.

P = probability
P(not A) = 1 − P(A)

1 List all outcomes

Each coin lands heads, H, or tails, T.

Here is every outcome.

Each coin has 2 ways.

2 times 2 times 2 is 8 outcomes, all just as likely.

all = 2 × 2 × 2 = 8

This list of all the outcomes is called the sample space.

H = heads, T = tails
sample space = all the outcomes

2 Find the opposite

Outcomes with no heads: just one.

That is 1 outcome.

no heads = 1

So the probability of no heads is 1/8.

P(no heads) = 18

3 Take it from 1

Now we take it from 1.

P(at least 1 head) = 1 − 18

Careful!

1 is all the outcomes: 8/8.

Careful: 1 = 88 (all of them)

So we write 1 as 8/8.

P(at least 1 head) = 88 − 18

Same bottoms: we take away the tops.

8 minus 1 is 7.

P(at least 1 head) = 78

4 Check

Let's check by counting the others straight away.

7 out of 8.

The same.

So the answer is right!

at least 1 head: 7 of 8 → 78

Let's look back.

First: list all outcomes.

Then: find the opposite.

Then: take it from 1.

Last: check.

Answer
P(at least 1 head) = 78

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. flip 2 coins, P(two heads)
    Answer
    Careful: not 13 ✗
  2. flip 2 coins, P(exactly 1 head)
    Answer
    P(exactly 1 head) = 12
  3. flip 2 coins, P(no heads)
    Answer
    Careful: not 13 ✗
  4. roll two dice, P(doubles)
    Answer
    P(doubles) = 16
  5. roll two dice, P(sum is 7)
    Answer
    P(sum 7) = 16
  6. roll two dice, P(sum is 2)
    Answer
    Careful: not 111 ✗

Medium

  1. flip 3 coins, P(exactly 2 heads)
    Answer
    Careful: not 14 ✗
  2. flip 3 coins, P(3 heads)
    Answer
    Careful: not 14 ✗
  3. flip 3 coins, P(exactly 1 head)
    Answer
    Careful: not 14 ✗
  4. roll two dice, P(sum is 8)
    Answer
    Careful: not 111 ✗
  5. roll two dice, P(sum more than 9)
    Answer
    P(sum more than 9) = 16
  6. roll two dice, P(sum less than 5)
    Answer
    P(sum less than 5) = 16
  7. roll two dice, P(sum is 11)
    Answer
    P(sum 11) = 118

Harder

  1. flip 3 coins, P(at least 1 head)
    Answer
    P(at least 1 head) = 78
  2. flip 3 coins, P(at least 2 heads)
    Answer
    P(at least 2 heads) = 12
  3. roll two dice, P(sum is not 7)
    Answer
    P(sum not 7) = 56
  4. roll two dice, P(not doubles)
    Answer
    P(not doubles) = 56
  5. flip 2 coins, P(at least 1 head)
    Answer
    P(at least 1 head) = 34
  6. roll two dice, P(sum more than 3)
    Answer
    P(sum more than 3) = 1112
  7. roll two dice, P(sum is 12)
    Answer
    Careful: not 111 ✗