Question

3. A deck of 40 cards consists of the following:

 10 black cards showing squares, numbered 1–10
 10 black cards showing circles, numbered 1–10
 10 red cards showing X’s, numbered 1–10
 10 red cards showing diamonds, numbered 1–10
A card will be selected at random from the deck.
a. i. Are the events "the card shows a square" and "the card is red" disjoint? Explain.
ii. Calculate the probability that the card will show a square or will be red.
b. i. Are the events "the card shows a 5" and "the card is red" disjoint? Explain.
ii. Calculate the probability that the card will show a 5 or will be red.

Answer 1

a) i. The events "the card shows a square" and "the card is red" are disjoint.

a) ii. Probability(square or red) =
`= (3)/(4)`

b) i. The events "the card shows a 5" and "the card is red" is not disjoint.

b) ii. Probability(card red or showing 5) =
`=(11)/(20)`

Explanation:

The total possible outcome is 40.

a) i. The events "the card shows a square" and "the card is red" are disjoint.

Events are said to be disjoint events when the events cannot happen at the same time. In other words, they are mutually exclusive.

So we the event "the card shows a square" and "the card is red" are disjoint because they cannot happen at the same time.

a) ii. Total number of square card = 10

Probability(square card) =
`(10)/(40)  = (1)/(4)`

Total number of red card = 20

Probability(red card) =
`(20)/(40)  = (2)/(4)  = (1)/(2)`

Probability(square or red) = probability(square) + probability(red)

Probability(square or red) =
`(1)/(4)  + (1)/(2) =(1 + 2)/(4)  = (3)/(4)`

b) i. The events "the card shows a 5" and "the card is red" is not disjoint.

The events are not disjoint because the two event can happen at the same time. We can have a card showing 5 and the card is red.

b) ii. Total number of red card = 20

Probability(red card) =
`(20)/(40)  = (2)/(4)  = (1)/(2)`

Total number of card showing 5 = 4

Probability(card showing 5) =
`(4)/(40)  =(1)/(10)`

Total number of card showing 5 and are red = 2

Probability(card showing 5 and is red) =
`(2)/(40)  = (1)/(20)`

Probability(card red or showing 5) = Prob(red card) + Prob(card show 5) - Prob(card show 5 and is red)

Probability(card red or showing 5) =

`=(1)/(2)+(1)/(10)-(1)/(20)\\=(5 + 1)/(10)-(1)/(20)\\=(6)/(10)-(1)/(20)\\=(12 - 1)/(20)\\=(11)/(20)`