Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
A compound event combines two or more simple ones: rolling two number cubes, flipping a coin and spinning a spinner, or choosing a sandwich and a drink. The key insight carries over from simple events. The probability is still the number of outcomes that make the event happen divided by the total number of equally likely outcomes in the sample space. What changes is that the sample space gets bigger, so students need reliable ways to list it.
Organized lists, two-way tables and tree diagrams each make sure no outcome is missed or counted twice. Rolling two number cubes gives a 6 by 6 table of 36 outcomes, and an event described in words, such as 'a total of 8', becomes a set of cells to count. Tree diagrams suit processes that happen in stages, with one branch for each outcome at each stage.
When listing everything is impractical, students design a simulation. Random digits, spinners or a computer stand in for the real process, and the results of many simulated trials estimate the probability. If 40% of blood donors have type A, the digits 0 to 3 can represent type A, and repeated strings of digits estimate how long it takes to find a type A donor.
Students often think a total of 2 is as likely as a total of 7. A table shows 1 way to make 2 but 6 ways to make 7.
With two distinguishable cubes these are different outcomes. Merging them shrinks the sample space and gives wrong probabilities.
For a coin and a spinner, P(heads and red) is not P(heads) + P(red). Counting outcomes in the sample space shows it is much smaller.
Two fair number cubes are rolled. What is the probability that the total is 8?
Answer: P(total of 8) = 5/36, about 0.14.
Start with a quick game: two players roll two cubes, and one scores on totals of 2, 3, 4, 10, 11 and 12, the other on 5 to 9. Students soon sense the game is unfair; building the 36-cell table explains why. Tree diagrams then generalize to processes with more stages or unequal numbers of outcomes.
Assessment asks students to draw or complete a sample space, count outcomes for an event described in words, or interpret the results of a simulation. Simulation items often ask students to explain how digits are assigned to outcomes, so practice writing that rule in a sentence.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 8
Each of the 2 coin results pairs with each of the 4 colors: 2 × 4 = 8 outcomes.
Answer: 1/36
Only (6, 6) works out of 36 equally likely outcomes, so P = 1/36.
Answer: D) 7
There are 6 ways to make 7, more than any other total. A total of 2 or 12 can each be made only 1 way, and 10 can be made 3 ways.
Answer: 1/12
There are 3 × 4 = 12 equally likely lunches and only one is tuna with orange juice, so P = 1/12.
Answer: B) 3/8
The 8 outcomes are HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Exactly two heads happens in HHT, HTH and THH, so P = 3/8.
Answer: C) 0, 1, 2, 3 = type A; 4 to 9 = not type A
Four of the ten digits is 40%, so four digits represent type A. The last option uses six digits (60%) for type A.
A full lesson with slides, activities and an exit ticket on probability of compound events, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.SP.C.8 with an answer key, ready in about a minute.
Make a worksheet →Turn probability of compound events into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Tables work neatly for two stages. Tree diagrams are better for three or more stages, or when the stages have different numbers of outcomes.
Not formally. They find compound probabilities by counting outcomes in a sample space. Multiplying probabilities is a natural extension later, in high school.