Geometric distribution

Each time a spinner is spun, the probability that it lands on red is 0.2

Find the probability that the spinner lands on red

for the 1st time on the 4th spin

for the 3rd time on the 8th spin

exactly 4 times during 10 spins

Each time the spinner is spun, the probability that it lands on yellow is 0.4

In a game with this spinner, a player must choose one of two events

R is the event that the spinner lands on red for the 1st time in at most 4 spins

Y is the event that the spinner lands on yellow for the 3rd time in at most 7 spins

Showing your calculations clearly, determine which of these events has the greater probability.

In a game a spinner is spun repeatedly. When the spinner is spun, the probability of it landing on blue is 0.11

Find the probability that the spinner lands on blue

for the first time on the 6th spin,

for the first time before the 6th spin,

exactly 4 times during the first 6 spins,

for the 4th time on or before the 6th spin.

Zac and Izana play the game. They take turns to spin the spinner. The winner is the first one to have the spinner land on blue. Izana spins the spinner first.

Show that the probability of Zac winning is 0.471 to 3 significant figures.

A random sample of 150 observations is taken from a geometric distribution with parameter 0.3

Estimate the probability that the mean of the sample is less than 3.45

Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red. Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.

The probability of Asha getting a red counter on any one draw is 0.07

Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.

Find the probability that Asha gets a red counter for the second time on her 9th draw.

The probability of Davinda getting a red counter on any one draw is p. Davinda draws counters until she gets n red counters. The random variable D is the number of counters Davinda draws.

Given that the mean and the standard deviation of D are 4400 and 660 respectively,

find the value of p.

Jerry believes that his bag contains a smaller proportion of red counters than Asha’s bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable J to be the number of counters drawn up to and including the first red counter.

Stating your hypotheses clearly and using a 10% level of significance, find the critical region for this test.

Jerry gets a red counter for the first time on his 34th draw.

Giving a reason for your answer, state whether or not there is evidence that Jerry’s bag contains a smaller proportion of red counters than Asha’s bag.

Given that the probability of Jerry getting a red counter on any one draw is 0.011

show that the power of the test is 0.702 to 3 significant figures.

Suzanne and Jon are playing a game.

They put 4 red counters and 1 blue counter in a bag.

Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.

Find the probability that Suzanne selects the blue counter on her 4th selection.

Find the probability that the blue counter is first selected on or after Jon’s third selection.

Find the mean and standard deviation of the number of selections made until the blue counter is selected.

Find the probability that Suzanne wins the game.