Lesson 15.1 · 20 min
Successive and Combined Events
Listing every outcome of two events, and multiplying probabilities when one event does not affect the other.
In this lesson you will learn to…
- Use the fact that the probabilities of all outcomes add up to 1
- Decide whether two events are independent or dependent
- List all the outcomes of two successive events in a sample space diagram
- Find the probability of one event and another happening
Why this matters
Almost nothing interesting in probability involves a single coin toss. Real questions stack events on top of each other: rain today and rain tomorrow, pass the first test and pass the second, pick a red counter then pick another.
This lesson gives you the two tools for handling that: a way of listing every outcome so nothing is missed, and a rule for multiplying when the events do not interfere with each other.
Listing outcomes without missing any
When two things happen one after the other, the trap is missing outcomes. A sample space diagram stops that happening because it is systematic.
There are two common forms:
- A two-way table — the first event down the side, the second across the top. Best when both events have a small number of outcomes.
- An ordered list — write every combination in a strict order, such as HH, HT, TH, TT.
Either way, the point is the same: every cell is one outcome, all outcomes are equally likely (if the coin and spinner are fair), and so
Independent or not?
Before you multiply, always ask one question: does the first event change the second?
| Situation | Independent? | Why |
|---|---|---|
| Roll a dice, then spin a spinner | Yes | The dice has no memory and no effect |
| Toss a coin twice | Yes | The coin does not know what it did last time |
| Take a sweet, eat it, take another | No | There is one fewer sweet in the bag |
| Take a card, put it back, take another | Yes | The pack is back to how it was |
That phrase “without replacement” is the giveaway for dependent events. If you see it, the second probability will have a different denominator from the first.
The two rules, side by side
It is easy to mix these up, so keep the key word in mind:
- AND → multiply. for independent events. Multiplying by a number less than 1 makes things smaller — which makes sense, because wanting two things to happen is harder than wanting one.
- OR → add. when A and B cannot both happen. Adding makes things bigger — also sensible, because you now have two ways to succeed.
A quick sanity check. Every probability you calculate must land between 0 and 1. If you get a number bigger than 1, you have almost certainly added when you should have multiplied.
Key rules and formulas
All the probabilities add to 1
If you list every outcome that could happen and none of them overlap, their probabilities must total exactly 1. So the probability something does not happen is .Independent events
Two events are independent when the first one happening does not change the chance of the second. Rolling a dice then spinning a spinner: independent. Taking two sweets out of a bag without putting the first one back: not independent, because the bag has changed.The AND rule
For independent events, multiply. This is the single most useful rule in the topic — but it only works when the events really are independent.The OR rule for outcomes that cannot both happen
If two outcomes are mutually exclusive — they cannot both happen at once — add their probabilities.
Spinner and outcomes
| Outcome | Red | Blue | Green | Yellow |
|---|---|---|---|---|
| Frequency | 0 | 0 | 0 | 0 |
| Relative frequency | — | — | — | — |
- Number of spins
- 0
- Experimental P(Red)
- —
- Theoretical P(Red)
- 1/4 = 0.25
- Difference
- —
Four equal sections, so each has probability . Spin it a few times — notice that the results are not evenly spread at first. That is normal, and the next lesson but one explains why.
Worked examples
Finding a missing probability
Try it yourself first, then reveal the solution one step at a time.
A sample space diagram
Try it yourself first, then reveal the solution one step at a time.
Two events in a row
Try it yourself first, then reveal the solution one step at a time.
Practice: combined events
9 questions
Question 1 of 9Type the answerFoundation A bag holds only red, blue and yellow beads. and . Find .Question 2 of 9True or false Nadia rolls a dice and then tosses a coin. These two events are independent.Question 3 of 9True or false A bag has 10 sweets. Jemma takes one and eats it, then takes another. These two events are independent.Question 4 of 9Multiple choice A fair coin is tossed twice. What is ?Question 5 of 9Complete the table A coin (H or T) is tossed and a three-sided spinner numbered 1, 2, 3 is spun. Complete the sample space diagram.Coin Spinner 1 Spinner 2 Spinner 3 H H1 T Question 6 of 9Type the answer Using that same coin and three-sided spinner, find .Question 7 of 9Choose all that apply Select every pair of events below that is independent.Select every correct answer — there may be more than one.
Question 8 of 9Fill the gaps Complete the rules by typing the missing words.For independent events, . For outcomes that cannot both happen, . All the probabilities of a complete set of outcomes add up to .
Question 9 of 9Step by step Work through this one step at a time.The probability that Arun cycles to school on any day is , and whether he cycles on one day has no effect on the next. Find the probability that he cycles on both Monday and Tuesday.- Are the two days independent?