Count Two-Draw Paths with Returns

List equally likely ordered tile pairs and distinguish one draw from two draws together.

All worksheets

Watch the lesson | Next: two draws without returns

5 questions0m 0s

Learn the skill

Give every physical tile its own name, even when shapes match. List the first tile, then the second. Returning the first tile restores every choice for the second draw. Count matching ordered pairs out of all allowed pairs. Exactly one star includes star then circle AND circle then star; two stars means both draws are stars. Reset the bag for each question.

Worked example

A separate worked bag has S1, C1 and C2. Return the first tile. Each of the 3 first tiles has 3 second choices, making 9 equally likely pairs. Only S1 then S1 gives two stars, so its chance is 1/9. S1 then C1 and C1 then S1 are different orders. The questions below use a different bag.

Worked bag S1, C1, C2 with returns. Nine ordered pairs fill a three-by-three grid. S1 then S1 is the only two-star pair. First tile is down the side; second is across the top. — Worked example only: one star and two circles. Use each question's own bag.

Would you believe it?

Two tiles can look the same but make different paths. Star tile S1 followed by S2 and S2 followed by S1 are two ordered results. Labels help count physical choices instead of assuming shape names are equally likely.

OpenStax: sample spaces and equally likely outcomes

Question 1 A NEW bag has S1, S2, S3 and C1. Each tile is equally likely. Draw twice, returning the first tile. How many equally likely ordered pairs are possible? 16 4 12

Question 2 Use S1, S2, S3, C1 with returns. What is the chance that BOTH draws are stars? 3/4 6/16 9/16

Question 3 Use the same four-tile bag with returns. What is the chance of EXACTLY ONE star in two draws, in either order? 3/16 6/16, or 3/8 9/16

Question 4 Use the same four-tile bag with returns. What is the chance of a star FIRST and a circle SECOND? 3/16 6/16 1/4

Question 5 Someone draws C1 then C1 from this bag, returning the first tile. Does that prove that two circles have probability 1? Yes; the observed pair must have been certain No; C1 then C1 is impossible No; one observed pair does not replace the model probability 1/16

Review your work

0correct

Complete the questions to review each answer.

View dashboard