# Separate events

Events related to a phenomenon are separate if the events do not have any common elementary cases. Example 1.

Are events A = “Take a card from a deck and it is a 6” and event B = “Take a card from a deck and it is a 7” separate?

Answer: Events A and B are separate because they do not have any common elementary cases.

Are events A = “Take a card from a deck and it is a 6 and event D = “Take a card from a deck and it is a spade” separate?

Answer: Events A and D are not separate events because they have one common elementary case, the 6 of spades. Example 2. There are 655 students in a high school, 210 of who exercise, 135 who play music, and 355 who do not exercise or play music. What is the probability that a high school student will both play music and exercise?

355 + 210 + 135 - 655 = 45 students play music and exercise

The information can be presented in a so-called Venn diagram  Answer: A student plays music and exercise with a probability of 0,069

### Rule of Multiplication for separate events

If we know that events A and B are independent events, then Example 3

Three cards are drawn from a deck and after each withdrawal the card is shuffled back into the deck.

a) Are the events separate?

b) What is the probability of getting three hearts in a row?

c) What is the probability of getting three aces in a row?

a) The events are separate because the card is always shuffled back into the deck, so previous draws of the card do not affect the probability of the following draws.  Example 4

Three cards are drawn from a deck and the cards are not shuffled back into the deck after the drawings.

a) Are the events separate

b) What is the probability of getting three hearts in a row?

c) What is the probability of getting three aces in a row?

a) The events are not separate, because drawing previous cards from the deck reduces the number of cards in the deck and so affects the probability of the following draws. Answer: The probability of getting three hearts in a row when the cards are not returned to the deck is 0,013. Answer: The probability of getting three aces in a row when the cards are not returned to the deck is 0,00018.

## Complementary events

The comlementary event of event A refers to all other elementary cases of some phenomenon than event A. The sum of the probabilities of event A and its complementary event is one, so Example 5

Write what is the complement of the event.

a) Event A = “roll a dice and get 6”

b) Event B = “roll a dice and not get more than 4”

c) Event C = “at least one student is late for class”

Solution

a) Counteraction = “get 5 or less”

b) Counteraction = “get 5 or more”

c) Countermeasure = “no student is late for class”

Example 6

Alma-Gunilla Heiskanen was on her way home from work. There are two bars on her way home, the 'Pub' and the 'Bistro'. Alma-Gunilla’s spouse Per-Ulf has noted her visits to the bar and ended up with the following separate probabilities. Alma-Gunilla has a 10% chance to visit the 'Pub' and a 20% chance to visit the 'Bistro'.

a) What is the probability that Alma-Gun will visit both bars on her way to home?

b) What is the probability that Alma-Gun will not visit either bar on her way home?

c) What is the probability that Alma-Gun will visit at least one bar on her way home?

It is known from the assignment that

P (“goes to the Pub”) = 0.10 P (“doesn't go to the Pub”) = 1 - 0.10 = 0.90

P (“goes to the Bistro”) = 0.20 P (“doesn’t go to the Bistro”) = 1 - 0.20 = 0.80

a) Visits both bars Answer: Alma-Gunilla will visit both bars with a probability of 2%.

b) Does not visit either Answer: Alma-Gunilla does not visit either bar with a probability of 72%.

c) The complement of the event “goes to at least one bar” is “does not go in either”, so Answer: Alma-Gunilla will visit at least one bar on her way home with a probability of 0,28.

Example 7

2% of the products sold by a company have a defective colour.

a) What is the probability that none of the ten products randomly selected has a defective colour?

b) What is the probability that at least one of the ten randomly selected products has a defective colour.

a) The events are separate Since P (“defective”) = 0,02, then P (“not defective”) = 0,98 and Answer: The probability of getting ten flawless products is 0,82

b) “At least one” questions are easiest to calculate using a complementary event. Answer: The probability that at least one of the ten products has a colour defective is 0,18.

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