Worksheet: Conditional Probability: Tree Diagrams

In this worksheet, we will practice using tree diagrams to calculate conditional probabilities.

Q1:

A bag contains 13 white balls and 11 black balls. If 2 balls are drawn consecutively without replacement, what is the probability that both balls are white?

  • A 1 3 2 4
  • B 1 3 4 6
  • C 1 2 2 3
  • D 1 6 9 5 7 6

Q2:

A bag contains 27 white balls and 6 black balls. If 2 balls are drawn consecutively without replacement, what is the probability that the second ball is black given that the first is black?

  • A 5 3 3
  • B 2 1 1
  • C 1 3 1 6
  • D 5 3 2

Q3:

A bag contains 8 red balls and 8 black balls. If two balls are drawn without replacement, what is the probability of getting one red ball and one black ball?

  • A 1 2
  • B 1 4
  • C 8 1 5
  • D 4 1 5

Q4:

The probability that it rains on a given day is 0.6. If it rains, the probability that a group of friends plays football is 0.2. If it does NOT rain, the probability that they play football rises to 0.8.

Work out the probability that it rains on a given day and the friends play football.

Work out the probability that it does NOT rain on a given day and the friends play football.

What is the probability that the friends will play football on a given day?

Q5:

A bag contains a total of nine marbles: three are blue and six are red. Michael randomly takes a marble from the bag, records its color, and puts it back in. He then repeats this process. He draws the following tree diagram.

Write the values of the probabilities 𝐴, 𝐡, and 𝐢 in the probability tree. Give your answers as unsimplified fractions.

  • A 𝐴 = 1 9 , 𝐡 = 1 9 , 𝐢 = 1 9
  • B 𝐴 = 6 9 , 𝐡 = 6 9 , 𝐢 = 6 9
  • C 𝐴 = 3 9 , 𝐡 = 3 9 , 𝐢 = 3 9
  • D 𝐴 = 1 2 7 , 𝐡 = 1 2 7 , 𝐢 = 1 2 7
  • E 𝐴 = 8 9 , 𝐡 = 8 9 , 𝐢 = 8 9

Use the probability tree to calculate the probability of choosing two blue marbles. Give your answer as a simplified fraction.

  • A 1 9
  • B 2 9
  • C 1 2 7
  • D 4 9
  • E 8 9

Calculate the probability of choosing at least one red marble. Give your answer as a simplified fraction.

  • A 2 3
  • B 5 9
  • C 1 9
  • D 8 9
  • E 4 9

Q6:

A bag contains 21 red balls and 15 black balls. If two balls are drawn without replacement, what is the probability of getting one red ball and one black ball?

  • A 3 5 7 2
  • B 3 5 1 4 4
  • C 1 2
  • D 1 4

Q7:

It is a little known fact that drugs have been used to enhance performance in sports since the original Olympic Games (776 to 393 BC). In fact, the origin of the word β€œdoping” is thought to come from the Dutch word β€œdoop,” which is a type of opium juice used by the ancient Greeks.

Currently, in all major sporting events, drug testing has become standard practice. However, it is known that not all those who test positive for drugs have actually taken drugs. This is called the β€œfalse positive effect.” Similarly, there is a β€œfalse negative effect,” where someone who tests negative for drugs has actually taken drugs.

In 2003, after anonymous testing of almost 1,500 players, the MLB (Major League Baseball) announced that approximately 6% of MLB players used performance-enhancing drugs. There was, however, a 5% chance that those who tested positive had not taken drugs and a 10% chance that those who had taken drugs tested negative.

Find the probability that an MLB player chosen at random had not taken drugs and tested positive.

Find the probability that an MLB player chosen at random had taken drugs and tested positive.

Find the probability that an MLB player chosen at random had positive test results.

Q8:

A bag contains 22 red balls and 15 black balls. One red ball is removed from the bag and then a second ball is drawn at random. Find the probability that the second ball is black. Give your answer to three significant figures.

Q9:

Suppose two spinners are spun. The first has 5 equal sectors numbered from 1 to 5, and the second has 9 equal sectors numbered from 1 to 9. Using a tree diagram or otherwise, find the probability that both spinners stop at odd numbers.

  • A30 out of 45
  • B12 out of 40
  • C27 out of 45
  • D25 out of 40
  • E15 out of 45

Q10:

A bag contains a total of 12 marbles; 4 are green and the rest are yellow. Madison randomly takes a marble from the bag, records its color, and does not replace it. She then repeats this process a second time. She draws the following tree diagram.

Write the values of the probabilities 𝐴, 𝐡, 𝐢, and 𝐷 in the probability tree. Give your answers as unsimplified fractions.

  • A 𝐴 = 4 1 2 , 𝐡 = 3 1 1 , 𝐢 = 8 1 1 , 𝐷 = 7 1 1
  • B 𝐴 = 4 1 2 , 𝐡 = 5 1 2 , 𝐢 = 7 1 2 , 𝐷 = 7 1 1
  • C 𝐴 = 4 1 2 , 𝐡 = 3 1 2 , 𝐢 = 9 1 2 , 𝐷 = 7 1 1
  • D 𝐴 = 4 1 2 , 𝐡 = 3 1 1 , 𝐢 = 8 1 1 , 𝐷 = 1 0 3 3
  • E 𝐴 = 4 1 2 , 𝐡 = 2 1 2 , 𝐢 = 2 1 2 , 𝐷 = 1 0 3 3

Calculate the probability of choosing two marbles that are of the same color. Give your answer as a simplified fraction.

  • A 1 1 1
  • B 1 4 3 3
  • C 1 6 3 3
  • D 1 7 3 3
  • E 1 9 3 3

Calculate the probability of choosing one marble of each color. Give your answer as a simplified fraction.

  • A 1 1 1
  • B 1 7 3 3
  • C 1 6 3 3
  • D 1 4 3 3
  • E 1 9 3 3

Q11:

A box contains 7 green balls and 5 red balls. If a ball is drawn at random and replaced and then another ball is drawn, find the probability that the first one is red and the second one is green.

  • A 1 2 1 1
  • B 1 4 4 3 5
  • C1
  • D 3 5 1 4 4
  • E 3 5 1 2 1

Q12:

A bag contains 9 purple balls and 6 blue balls. If two balls are drawn one after another without replacement, find the probability that one of them is blue and the other one is purple.

  • A 7 0 9
  • B 1 8 3 5
  • C 3 4 2 5
  • D 9 9 7 0
  • E 1 2 2 5

Q13:

If a coin is flipped and then a die is thrown, what is the probability of the coin landing on heads and the die landing on number 3?

  • A 2 3
  • B 1 3
  • C 5 6
  • D 1 1 2
  • E 1 6

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