Spinner With 6 Zones 3 Red to Green 1 Blue Spun 3 Times Odds It Lands in Green All 3 Times

Extending Probability

Multiplication for Independent Events

Two events are independent if one event happening does not affect the probability of the other event. In this case the probability of two events A and B occurring is given by

p(A and B) = p(A) × p(B)

Worked Examples

A die is rolled twice. If event A is the first roll shows a six and event B is the second roll shows a six,

(a)

are events A and B independent?

Show me

The events are independent as the number obtained on the first roll does not affect the second roll.

(b)

Find p(A and B).

Show me

	p(A)	= and p(B) =
so	p(A and B)	= p(A) × p(B)
		= ×
		=

A spinner in game has 3 sections of equal size that are coloured red, blue and green. Let the events B, R and G be:

B: the spinner lands a blue

G: the spinner lands on green

R: the spinner lands on red.

(a)

Are these events independent?

Show me

The events are independent as the result of one spin does not affect the next.

(b)

Find the probability that when the spinner is spun twice the following outcomes are obtained.

The probabilities of each event are

p(B) = p(R) = p(G) = .

(i)

Red both times.

Show me

The probability is given by

p(R and R)	= p(R) × p(R)
	= ×
	=

(ii)

Red and green in any order.

Show me

For red and green in any order, two outcomes must be considered: Red then Green and Green then Red.

p(G and R)	= p(G) × p(R)
	= ×
	=

p(R and G)	= p(R) × p(G)
	= ×
	=

Hence the probability of a red and green in any order is given by:

p(R and G) + p(G and R)	= +
	=

(iii)

Both are the same colour.

Show me

For both to be the same colour the outcomes, R and R, G and G, B and B must be considered.

From (i) p(R and R) =

Similarly p(B and B) =

and p(G and G) =

The probability that both spins are the same colour is given by:

p(R and R) + p(B and B) + p(G and G)	= + +
	=
	=

Exercises

For each pair of events A and B listed below, decide whether or not it is likely that the events are independent.

(a)

A: It rains today.

B: It rains tomorrow.

(b)

A: It rains on Monday this week.

B: It rains on Monday next week.

(c)

A die is rolled twice.

A: The first roll shows a 3.

B: The second roll shows a 5.

(d)

A baby is born.

A: Its left eye is blue.

B: Its right eye is blue.

(e)

Joshua and James are brothers.

A: Joshua catches measles.

B: James catches measles.

(f)

Daniel cycles to school.

A: Daniel's bicycle has a puncture.

B: Daniel is late for school.

The spinner shown in the diagram has eight sections of equal size; each one is coloured white or black.

The events B and W are:

B: the spinner lands on black,

W: the spinner lands on white.

(a)

Find the following probabilities:

(i)

p(B)

(ii)

p(W)

(iii)

p(B and B)

(iv)

p(W and W)

(v)

p(B and W)

(vi)

p(W and B).

(b)

If the spinner is spun twice find the probabilities of the following outcomes.

(i)

White is obtained both times.

(ii)

A different colour is obtained on each spin.

(iii)

The same colour is obtained on each spin.

A bag contains 7 red balls and 3 green balls. A ball is taken out and replaced. A second ball is then taken out.

R is the event that a Red ball is selected. G is the event that a Green ball is selected.

(a)

Find the following probabilities

(i)

p(R)

(ii)

p(G)

(iii)

p(G and G)

(iv)

p(R and R)

(v)

p(G and R)

(vi)

p(R and G).

(b)

Find the probability that if two balls are taken in turn;

(i)

they are both red,

(ii)

they are different colours,

(iii)

they are the same colour.

In the game of 'Pass the Pig', two identical toy pigs are thrown. Each pig can land in one of five positions. The five positions and the probabilities that the pig will land in each of these positions are shown in the table.