Conditional and Independent Probability
Finding probabilities with dependent and independent events
Conditional and Independent Probability
Finding probabilities with dependent and independent events
Math - Grade 9-12
- 1
A bag contains 5 red marbles and 7 blue marbles. One marble is drawn at random and not replaced. Given that the first marble drawn was red, what is the probability that the second marble is blue?
Use the condition that one red marble has already been removed.
If the first marble was red, then 11 marbles remain and 7 of them are blue. The probability that the second marble is blue is 7/11. - 2
A coin is flipped and a number cube labeled 1 through 6 is rolled. What is the probability of getting heads and an even number?
The probability of heads is 1/2 and the probability of an even number is 3/6, or 1/2. Because these events are independent, multiply: 1/2 x 1/2 = 1/4. The probability is 1/4. - 3
In a class of 30 students, 18 play a sport, 12 are in the school band, and 8 do both. What is the probability that a randomly chosen student is in band given that the student plays a sport?
Use P(Band | Sport) = P(Band and Sport) / P(Sport).
Conditional probability is the number in both groups divided by the number in the given group. Since 8 students are in band and play a sport, and 18 play a sport, the probability is 8/18, which simplifies to 4/9. - 4
A jar has 4 green tiles, 3 yellow tiles, and 5 black tiles. Two tiles are chosen at random without replacement. What is the probability that both tiles are green?
The probability of green on the first draw is 4/12, or 1/3. Then 3 green tiles remain out of 11 total tiles, so the second probability is 3/11. Multiply to get 1/3 x 3/11 = 1/11. The probability is 1/11. - 5
A survey found that 40% of students have a part-time job and 25% of students participate in theater. If having a part-time job and participating in theater are independent, what is the probability that a student does both?
Independent events use multiplication.
For independent events, multiply the probabilities. So 0.40 x 0.25 = 0.10. The probability that a student does both is 0.10, or 10%. - 6
A standard deck of 52 cards is used. One card is drawn, not replaced, and then a second card is drawn. What is the probability that the second card is a heart given that the first card was a heart?
If the first card was a heart, then 12 hearts remain in a deck of 51 cards. The probability that the second card is a heart is 12/51, which simplifies to 4/17. - 7
A spinner has 5 equal sections labeled A, B, C, D, and E. The spinner is spun twice. Are the events 'getting A on the first spin' and 'getting A on the second spin' independent or dependent? Explain.
Ask whether the first event changes the second event.
These events are independent because the result of the first spin does not change the spinner or affect the result of the second spin. Each spin still has a 1/5 chance of landing on A. - 8
At a school, 60 students take Spanish, 45 take art, and 20 take both out of 100 students total. What is the probability that a randomly chosen student takes art given that the student takes Spanish?
The conditional probability is the number who take both divided by the number who take Spanish. That is 20/60, which simplifies to 1/3. The probability is 1/3. - 9
Two cards are drawn from a standard deck without replacement. What is the probability that the first card is a king and the second card is a queen?
Because there is no replacement, the second probability uses 51 cards.
The probability the first card is a king is 4/52, or 1/13. Then 4 queens remain out of 51 cards, so the probability of a queen second is 4/51. Multiply to get 1/13 x 4/51 = 4/663. The probability is 4/663. - 10
A box contains 9 math books and 6 science books. One book is selected, replaced, and then a second book is selected. What is the probability of choosing a science book both times?
Because the first book is replaced, the events are independent. The probability of choosing a science book each time is 6/15, or 2/5. Multiply: 2/5 x 2/5 = 4/25. The probability is 4/25. - 11
A restaurant found that 70% of customers order a drink and 30% order dessert. If 21% of customers order both, are ordering a drink and ordering dessert independent? Explain.
Compare P(Drink and Dessert) to P(Drink) x P(Dessert).
If the events were independent, the probability of both would be 0.70 x 0.30 = 0.21. Since this matches the given probability of both, the events are independent. - 12
In a drawer there are 8 white socks and 4 black socks. Two socks are chosen at random without replacement. What is the probability that the second sock is black given that the first sock was white?
If the first sock was white, then 11 socks remain and 4 of them are black. The conditional probability that the second sock is black is 4/11. - 13
A student answers a true-false question and then rolls a fair six-sided die. What is the probability of answering the question correctly by guessing and rolling a number greater than 4?
Numbers greater than 4 on a die are 5 and 6.
The probability of guessing a true-false question correctly is 1/2. The probability of rolling a number greater than 4 is 2/6, or 1/3. These events are independent, so multiply: 1/2 x 1/3 = 1/6. The probability is 1/6. - 14
A club has 24 members. Fifteen members are juniors, and 9 of those juniors are also on the debate team. What is the probability that a randomly selected club member is on the debate team given that the member is a junior?
Given that the member is a junior, the sample space is the 15 juniors. Of those, 9 are on the debate team. So the conditional probability is 9/15, which simplifies to 3/5. - 15
A bag contains 3 orange counters and 2 purple counters. Two counters are drawn without replacement. Find the probability of drawing an orange counter first and a purple counter second.
Multiply the probability of the first draw by the probability of the second draw after the first counter is removed.
The probability of drawing orange first is 3/5. Then 2 purple counters remain out of 4 total counters, so the probability of purple second is 2/4, or 1/2. Multiply: 3/5 x 1/2 = 3/10. The probability is 3/10.