Two-Stage Probability Tree Diagrams Worksheet

A coin is flipped twice. What is the probability of getting Heads on both flips?

Multiply along the Heads-Heads path: (1/2) * (1/2) = 1/4.

A bag holds 3 red marbles and 2 blue marbles. One marble is drawn, noted, and replaced before drawing a second. What is the probability of drawing two red marbles?

With replacement, denominators stay 5: (3/5) * (3/5) = 9/25.

From the same bag (3 red, 2 blue), two marbles are drawn without replacement. What is the probability that both marbles are red?

Without replacement, the second draw drops to 4: (3/5) * (2/4) = 6/20, which simplifies to 3/10.

Two marbles are drawn without replacement from a bag of 3 red and 2 blue marbles. What is the probability of drawing one red and one blue marble in any order?

Add the two valid paths: Red then Blue is (3/5) * (2/4) = 6/20. Blue then Red is (2/5) * (3/4) = 6/20. Total: 6/20 + 6/20 = 12/20 = 3/5.

A spinner has 3 equal sections: Red, Green, and Yellow. It is spun twice. What is the probability of spinning Green at least once?

Add the paths with Green: (Green, Green) = 1/9; (Green, Other) = 2/9; (Other, Green) = 2/9. Sum: 1/9 + 2/9 + 2/9 = 5/9. Alternatively, 1 - (2/3 * 2/3) = 5/9.

A drawer contains 4 black socks and 2 white socks. You take two socks without replacement in the dark. What is the probability of picking a matching pair?

Add both matching paths: Black-Black is (4/6) * (3/5) = 12/30. White-White is (2/6) * (1/5) = 2/30. Sum: 12/30 + 2/30 = 14/30 = 7/15.