Worksheet: Independent Events

In this worksheet, we will practice calculating probabilities for independent events.

Q1:

If 𝑃(𝐴)=0.3 and 𝑃(𝐡)=0.25 and 𝐴∩𝐡=βˆ…, are 𝐴 and 𝐡 independent?

  • AYes, they are independent.
  • BNo, they are dependent.

Q2:

What is the probability of getting tails at least once if a coin is flipped three times?

  • A38
  • B12
  • C78
  • D18

Q3:

In a sample space 𝑆, the probabilities are shown for the combinations of events 𝐴 and 𝐡 occurring. Are 𝐴 and 𝐡 independent events?

  • Ayes
  • Bno

Q4:

Three friends were all born in the same year, which was not a leap year. Assuming that each friend’s birthday is independent of the others’ birthdays, and that every day of the year is equally likely to be a birthday, find the probability that the friends all have the same birthday.

  • A3365
  • B13
  • C148,627,125
  • D1365
  • E1133,225

Q5:

David and Olivia applied for life insurance. The company has estimated that the probability that David will live to be at least 85 years old is 0.6 and the probability that Olivia will live to be at least 85 years old is 0.25. Given that these are independent events, what is the probability they will both live to be at least 85?

Q6:

In the given sample space 𝑆, the probabilities are shown for combinations of events 𝐴 and 𝐡 occurring. Are 𝐴 and 𝐡 independent events?

  • Ayes
  • Bno

Q7:

What is the probability of tossing 3 coins and getting tails on all three?

  • A18
  • B14
  • C19
  • D12
  • E132

Q8:

In which of the following scenarios are 𝐴 and 𝐡 independent events?

  • AA die is rolled and a coin is flipped. Event 𝐴 is rolling a 6 on the die, and event 𝐡 is the coin landing with its heads side up.
  • BA student leaves their house on their way to school. Event 𝐴 is them arriving at the bus stop in time to catch the bus and event 𝐡 is them getting to school on time.
  • CA child takes two candies at random from a bag which contains chewy candies and crunchy candies. Event 𝐴 is them taking a chewy candy first and event 𝐡 is them taking a crunchy candy second.
  • DA teacher selects two students at random from a group containing five boys and five girls. Event 𝐴 is the teacher selecting a boy first, and event 𝐡 is the teacher selecting a girl second.
  • EA coin is flipped. Event 𝐴 is rolling an even number and event 𝐡 is rolling a prime number.

Q9:

In which of the following scenarios are 𝐴 and 𝐡 independent events?

  • AA bag contains strawberry and chocolate candies, where event 𝐴 is randomly picking a first candy from the bag, eating it, and it being chocolate and event 𝐡 is randomly picking a second candy from the bag, eating it, and it being chocolate.
  • BA class of students contains eighteen boys and seventeen girls, and a teacher chooses two students at random, where event 𝐴 is the first randomly chosen student being a boy and event 𝐡 is the second randomly chosen student being a girl.
  • CEvent 𝐴 is a cat landing on its feet when blown from a branch of a tree by a gust of wind, while event 𝐡 is a piece of toast landing with its buttered side up on the floor when accidentally knocked from a breakfast plate on the table.
  • DOn a cold winter’s day, a student walks to school, where event 𝐴 is the student remembering to take his winter jacket and event 𝐡 is him arriving at school without getting too cold.
  • EA fair six-sided die is rolled, where event 𝐴 is getting an odd number and event 𝐡 is getting a prime number.

Q10:

In the given sample space 𝑆, the probabilities are shown for combinations of events 𝐴 and 𝐡 occurring. Are 𝐴 and 𝐡 independent events?

  • Ano
  • Byes

Q11:

Which of the following are two independent events?

  • ADrawing a king from a deck of 52 cards without replacement and then drawing an ace.
  • BDrawing a king from a deck of 52 cards and an ace from another well-shuffled deck of 52 cards.
  • CDrawing an ace twice from a well-shuffled deck of 52 cards.
  • DAll of the above.

Q12:

A jar of marbles contains 4 blue marbles, 5 red marbles, 1 green marble, and 2 black marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. Find the probability that the first is blue and the second is red.

  • A23
  • B0
  • C13
  • D34
  • E536

Q13:

Suppose 𝐴 and 𝐡 are two events in a random experiment. Given that 𝑃(𝐴βˆͺ𝐡)=0.68, 𝑃(𝐴∩𝐡)=0.12, and 𝑃(𝐴∩𝐡)=𝑃(𝐴)×𝑃(𝐡), what are the possible values for 𝑃(𝐴) and 𝑃(𝐡)?

  • A𝑃(𝐴)=0.4, 𝑃(𝐡)=0.2 or 𝑃(𝐴)=0.2, 𝑃(𝐡)=0.4
  • B𝑃(𝐴)=0.6, 𝑃(𝐡)=0.2 or 𝑃(𝐴)=0.2, 𝑃(𝐡)=0.6
  • C𝑃(𝐴)=0.6, 𝑃(𝐡)=0.8 or 𝑃(𝐴)=0.8, 𝑃(𝐡)=0.6
  • D𝑃(𝐴)=0.4, 𝑃(𝐡)=0.3 or 𝑃(𝐴)=0.3, 𝑃(𝐡)=0.4
  • E𝑃(𝐴)=0.4, 𝑃(𝐡)=0.8 or 𝑃(𝐴)=0.8, 𝑃(𝐡)=0.4

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