Grade 7 · Build a two-stage outcome table · 230 of 750
Build a two-stage outcome table · practice 17
Two independent selections each have six equally likely outcomes. The first has 5 favorable outcomes and the second has 5. Find the probability both are favorable. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Two independent selections each have six equally likely outcomes. The first has 5 favorable outcomes and the second has 5. Find the probability both are favorable.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Count the ordered pairs, then the favorable rectangle in the outcome grid.
Work through the reasoning
Step 1
Translate the givens
Independence gives 6 × 6 = 36 equally likely ordered pairs.
Step 2
Calculate with the model
Each of the 5 favorable first outcomes can pair with 5 favorable second outcomes: 5 × 5 = 25.
Step 3
Check and interpret
P(both) = 25/36 = (5/6)(5/6) ≈ 69.44444%. This multiplication depends on independence.
The answer
25/36 ≈ 69.44444%.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
Two independent six-outcome selections create a 6-by-6 grid of equally likely ordered pairs. If a outcomes are favorable in the first selection and b in the second, the successful rectangle contains ab pairs. Independence is what permits multiplication of the two probabilities. Starting quantities: Favorable first outcomes = 5; Favorable second outcomes = 5.
P(both) = (a/6)(b/6) = ab/36
03 · Reflect and transfer
Explain what changes and why.
What extra information would be needed for selections made without replacement?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.