Worksheet: Conditional Probability: Tree Diagrams

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

Q1:

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?

  • A533
  • B211
  • C1316
  • D532

Q2:

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?

Q3:

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 “doping” is thought to come from the Dutch word “doop,” which is a type of opium juice used by the ancient Greeks.

Drug testing has become standard practice. 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. They got this result taking into account that there was a 5%‎ chance that those who tested positive had not taken drugs and a 10%‎ chance that those who had taken drugs tested negative (false-negative effect).

Find the probability that an MLB player chosen at random had not taken drugs and tested positive. Round your answer to three decimal places if necessary.

Find the probability that an MLB player chosen at random had taken drugs and tested positive. Round your answer to three decimal places if necessary.

Find the probability that an MLB player chosen at random had positive test results. Round your answer to three decimal places if necessary.

Q4:

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 decimal places.

Q5:

A bag contains three red marbles, two yellow marbles, and six blue marbles. You take a marble at random from the bag and record its color. Then, without replacing the first marble, you take a second marble from the bag and record its color.

Given that the first marble you take is red, what is the probability that the second marble you take is also red?

  • A15
  • B355
  • C211
  • D2655
  • E311

What is the probability that the second marble you take is red, regardless of the color of the first marble you take?

  • A310
  • B655
  • C311
  • D15
  • E355

What is the probability that you take at least one red marble?

  • A13
  • B311
  • C15
  • D355
  • E2755

Q6:

A bag contains 4 red balls and 3 blue balls. I take one at random, note its color, and put it on a shelf. I then take another ball at random, note its color, and put it on the shelf next to the first ball. The figure below shows the probability tree associated with this problem. Are the events of “getting a blue ball on the first draw” and “getting a red ball on the second draw” independent?

  • Ano
  • Byes

Q7:

Jacob flips a coin and then rolls a six-sided die. He draws a tree diagram to represent this.

Find the probability that the die showed a number less than 3 given that the coin showed tails.

  • A12
  • B14
  • C23
  • D13
  • E16

Q8:

Jennifer travels to school by car or on foot.

The probability that she travels by car is 0.4 and the probability that she walks is 0.6.

If she travels by car, the probability that she is late is 0.2; if she travels on foot, the probability that she is late is 0.3. Using a tree diagram, calculate the probability that she is late given that she traveled by car.

Q9:

A bag contains 3 red marbles and 5 blue marbles. Two marbles are selected without replacement. Using a tree diagram, find the probability that the second marble is red, given that the first one is red. Give your answer to the nearest two decimal places.

Q10:

A bag contains 2 black balls and 8 white balls. Isabella selects two balls without replacement and draws the following tree diagram.

Find the values of 𝑎 and 𝑏.

  • A𝑎=19 and 𝑏=79
  • B𝑎=19 and 𝑏=89
  • C𝑎=29 and 𝑏=79
  • D𝑎=79 and 𝑏=19
  • E𝑎=110 and 𝑏=710

Hence, calculate the probability that the second ball is white, given that the first ball is white.

  • A19
  • B45
  • C710
  • D89
  • E79

Q11:

Two cards are drawn from an ordinary 52-card deck without replacement.

Find the probability that the second card drawn is a heart given that the first is not a heart.

  • A3952
  • B1351
  • C3951
  • D1352
  • E1251

Find the probability that both cards are hearts.

  • A1617
  • B1352
  • C117
  • D116
  • E1251

Find the probability that neither of the two cards is a heart.

  • A1934
  • B3952
  • C916
  • D152153
  • E3851

Find the probability that the first card drawn is a heart and the second one is not a heart.

  • A3951
  • B1752
  • C1351
  • D1368
  • E52153

Find the probability that one of the cards drawn is a heart and the other is not.

  • A1351
  • B169357
  • C1334
  • D5152
  • E3951

If we draw a third card, find the probability that it is a heart given that the first two cards are hearts.

  • A217
  • B1150
  • C164
  • D1352
  • E1152

Q12:

A bag contains 3 pink marbles, 4 orange marbles, and 5 yellow marbles. Two marbles are selected without replacement. Using a tree diagram, find the probability that the second marble is yellow given that the first marble is not yellow.

  • A712
  • B411
  • C511
  • D711
  • E611

Q13:

A bag contains 3 blue balls and 7 red balls. David selects 2 balls without replacement and draws the following tree diagram.

Given that the first ball is red, find the value of 𝑥 that represents the probability that the second ball selected is red.

  • A13
  • B79
  • C710
  • D23
  • E35

Q14:

There are three identical boxes numbered from 1 to 3. Each box contains 7 light bulbs. Box 𝑖 has 𝑖 defective bulbs and 7𝑖 nondefective bulbs, where 𝑖=1,2,3. For instance, box 2 contains 2 defective bulbs and 72=5 nondefective bulbs. We first select a box at random, then we select a light bulb from that box.

Use a tree diagram to find the probability of getting a defective bulb if we have already selected box 3. Round your answer to 3 decimal places.

Use a tree diagram to find the probability of getting a defective bulb from box 2. Round your answer to 3 decimal places.

Use a tree diagram to find the probability of getting a nondefective bulb. Round your answer to 3 decimal places.

Q15:

You have two boxes. Box 1 contains 5 red, 7 black, and 6 white balls. Box 2 contains 6 red, 5 black, and 4 white balls. We roll a fair die once. If we get 1 or 2, we will select one ball from box 1. If we get 3 or 4, we will select one ball from box 2. If we get 5 or 6, we will select a ball from box 1 and a ball from box 2.

Find the probability of selecting exactly one red ball. Round your answer to three decimal places.

Find the probability of selecting at least one red ball. Round your answer to three decimal places.

Q16:

Two students are selected at random from a class of 6 male and 4 female students.

Using a tree diagram, find the probability that both selected students are females. Round your answer to three decimal places.

Using a tree diagram, find the probability that exactly one of the selected students is a male. Round your answer to three decimal places.

Q17:

An ice cream store sells only two flavors, chocolate and vanilla. 60% of the sales are for the chocolate flavor. The ice cream is sold in cones or cups. The percentage of sales of ice cream cups for the chocolate flavor is 5%, and the percentage of sales of ice cream cones for the vanilla flavor is 55%.

For a randomly selected sale, find the probability of the following.

The ice cream is sold in a cup, given that it is vanilla flavored.

The ice cream is sold in a cup and it is vanilla flavored.

The ice cream is sold in a cone, given that it is chocolate flavored.

The ice cream is sold in a cone and it is chocolate flavored.

Q18:

You have two boxes. Box 1 contains 15 red balls and 3 black balls. Box 2 contains 5 red balls and 6 black balls. A ball is selected at random from box 1 and transferred to box 2. A ball is then selected at random from box 2.

If the ball selected from box 1 is red, find the probability of selecting a black ball from box 2.

  • A512
  • B35
  • C611
  • D712
  • E12

If the ball selected from box 1 is black, find the probability of selecting a black ball from box 2.

  • A772
  • B712
  • C12
  • D112
  • E611

What is the probability that the ball selected from box 2 is black, regardless of the color of the ball selected from box 1?

  • A712
  • B12
  • C112
  • D3772
  • E611

Q19:

Consider a group of 300 patients. 15 of them have normal blood pressure and the rest have high blood pressure. 20% of those who have high blood pressure are overweight.

If we choose a patient and detect that they have high blood pressure, find the probability that they are not overweight.

For a selected patient, find the probability that they are overweight and that they have high blood pressure.

Q20:

A truth serum works with 86% efficiency if the person is telling the truth and 98% efficiency if the person is lying. If a person is selected from a group of people of whom 30% are telling the truth, use a tree diagram to find the probability of the following.

The serum indicates that this person is a liar given that they are telling the truth.

This person is a liar and the serum indicates that they are telling the truth.

Q21:

In a call center, calls are classified into two types: a voice call, if someone is speaking, and a data call, if the call is a fax or a recorded message. Two employees are responsible for responding to these calls. The probability that the first employee responds is 0.4. If the second employee responds, the probability of the call being a data call is 0.61. Find the probability that a voice call is received and the second employee responds.

Q22:

You have two boxes. Box 1 contains 3 red balls and 5 black balls. Box 2 contains 4 red balls and 15 black balls. A biased coin with a probability of 23 of coming up heads is tossed. If it comes up heads, you will draw a ball from box 1. If it comes up tails, you will draw a ball from box 2.

Use a tree diagram to find the probability of getting a black ball from box 2. Round your answer to 3 decimal places if needed.

Use a tree diagram to find the probability of getting a black ball. Round your answer to 3 decimal places.

Use a tree diagram to find the probability of getting a red ball. Round your answer to 3 decimal places.

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