Topic 4.6: Combined Events Questions

A fair die is rolled and a fair coin is tossed. Find the probability of a six and a head.

Events A and B are independent with P(A) = 0.7 and P(B) = 0.2. Find P(A ∩ B).

A spinner with probability 1/4 of landing on red is spun once. If it lands on red, a card is taken from Pile A; otherwise a card is taken from Pile B. Pile A holds 2 white cards and 3 red cards; Pile B holds 1 white card and 4 red cards. The tree diagram shows the two stages, with the Pile B probabilities marked p and q.

Write down the value of p and the value of q.

A proportion $p$ of a population has a condition. A screening test is positive for $92\%$ of people with the condition and for $6\%$ of people without it. Draw a tree diagram to show this information, labelling every branch with its probability.

$P(A)=0.45,P(B)=0.35$ and $P(A\cap B)=0.15$. Find $P(A\cup B)$.

A spinner with probability 1/4 of landing on red is spun once. If it lands on red, a card is taken from Pile A; otherwise a card is taken from Pile B. Pile A holds 2 white cards and 3 red cards; Pile B holds 1 white card and 4 red cards. The tree diagram shows the two stages, with the Pile B probabilities marked p and q. For Pile B, p = 1/5 and q = 4/5.

Show that the probability of drawing a white ball is 1/4.

Events A and B are mutually exclusive with P(A) = 0.28 and P(B) = 0.34. Find P(A ∪ B) and P(A ∩ B).

In a school of 200 students, 110 play football (F), 80 play tennis (T) and 40 play neither, so 30 play both.

Show that the events F and T are not independent.

The probability of rain on Monday is $0.3$. If it rains on Monday, the probability of rain on Tuesday is $0.6$. If it does not rain on Monday, the probability of rain on Tuesday is $0.2$.

Draw a tree diagram to show this information, labelling every branch with its probability.

Draw a tree diagram for this situation.

A game has two independent events: $A$ is winning a raffle, with $P(A)=0.6$, and $B$ is winning a prize draw, with $P(B)=0.35$. Find $P(A\cap B)$.

Write down $P(\text{not win})$ on that branch.

A and B are mutually exclusive with $P(A)=0.32$ and $P(B)=0.41$. Find $P(A\cup B)$.

Show that $P(A\cap B)=0.28$.

A fair coin is tossed and a fair four-sided die, numbered $1$ to $4$, is rolled. Write down the probability of getting a head and a $4$.

A game is played by tossing a biased coin and then drawing a ball from one of two bags. If the coin shows heads a ball is drawn from Bag A, otherwise from Bag B. The probability of heads is 1/4. Bag A contains 2 white balls and 3 red balls. Bag B contains 1 white ball and 4 red balls. This is shown in the tree diagram, where the probabilities p and q are for Bag B. For Bag B, p = 1/5 and q = 4/5.

Show that the probability of drawing a white ball is 1/4.

In a school of 80 students, 44 play art (A), 30 play biology (B) and 16 play neither, so 10 play both.

Show that the events A and B are not independent.

A game is played by tossing a biased coin and then drawing a ball from one of two bags. If the coin shows heads a ball is drawn from Bag A, otherwise from Bag B. The probability of heads is 1/3. Bag A contains 1 white ball and 5 red balls. Bag B contains 3 white balls and 5 red balls. This is shown in the tree diagram, where the probabilities p and q are for Bag B. For Bag B, p = 3/8 and q = 5/8.

Show that the probability of drawing a white ball is 11/36.

In each round of a game, a player wins a token with probability $\frac{1}{3}$, independently of other rounds. Show that the probability that the player has exactly one token after two rounds is $\frac{4}{9}$.

Each day, independently, a bus is late with probability $0.2$. Write down the probability that the bus is late on Monday and on time on Tuesday.