Mutually exclusive and independent events

Sam has 9 cards, each with a different non-zero digit printed on it.

Sam chooses 5 cards at random and places them in a random order in a straight line, to form a 5-digit number.

Find the probability that the 5-digit number is greater than 50 000.

Given that the 5-digit number is greater than 50 000, determine the probability that it is

greater than 58 600.

The probability distribution of the random variable X is modelled as follows.

• P(X = 1) = p where p is a constant.

• P(X = x) = 2P(X = x - 1) for x = 2, 3, 4.

• P(X = x) = 0 for all other values of x.

Three values, X₁, X₂ and X₃, of X are chosen at random.

Determine P(X₃ > X₁ + X₂).

It is given that any integer can be expressed in the form 3 m + r, where m is an integer and r is 0, 1 or 2.

Use this fact to answer the following.

By considering the different values of r, prove that the square of any integer cannot be

expressed in the form 3 n + 2, where n is an integer.

Three integers are chosen at random from the integers 1 to 99 inclusive. The three integers are not necessarily different.

By considering the different values of r, determine the probability that the sum of these three

integers is divisible by 3.

Anika and Beth are playing a game which consists of several points.

• The probability that Anika will win any point is 0.7.

• The probability that Beth will win any point is 0.3.

• The outcome of each point is independent of the outcome of every other point.

The first player to win two points wins the game.

Write down the probability that the game consists of more than three points.

Complete the probability tree diagram in the Printed Answer Booklet showing all the

possibilities for the game.

Determine the probability that Beth wins the game.

Determine the probability that the game consists of exactly three points.

Given that Beth wins the game, determine the probability that the game consists of exactly

three points.

The probability distribution of a random variable X is modelled as follows.

where k is a constant.

Show in a table the values of X and their probabilities.

The values of three independent observations of X are denoted by X₁, X₂ and X₃.

Find P(X₁ > X₂ + X₃).

In a game, a player notes the values of successive independent observations of X and keeps a running total. The aim of the game is to reach a total of exactly 7.

Determine the probability that a total of exactly 7 is first reached on the 5th observation.

Andy and Bev are playing a game.

• The game consists of three points.

• On each point, P(Andy wins) = 0.4 and P(Bev wins) = 0.6.

• If one player wins two consecutive points, then they win the game, otherwise neither player wins.

Determine the probability of the following events.

Andy wins the game.

Neither player wins the game.

Andy and Bev now decide to play a match which consists of a series of games.

• In each game, if a player wins the game then they win the match.

• If neither player wins the game then the players play another game.

Determine the probability that Andy wins the match.

Find the probability that S is odd.

Find the probability that S is greater than 8, given that S is odd.

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable Y defined as follows. P(Y = y+1) = (Y = y) for all positive integers y.

Find P(Y = 1).

Give a reason why one of the variables, X or Y, might be more appropriate as a model for the number

of attempts that Sheila needs to start her car.