Probability

What is Probability

• We using the idea of probability to describe How likely something is to happen

Example-Tossing a Coin

• When a coin is tossed, there are two possible outcomes:

• heads (H) or • tails (T)• We say that the probability of the coin landing

H is ½. • And the probability of the coin landing T is ½.

Example-Throwing Dice

• When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

• The probability of any one of them is 1/6.

Probability

• Probability of an event happening = Number of ways it can happen

Total number of outcomes

Example

• There are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Example

• There are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

• Number of ways it can happen: 4 (there are 4 blues)

• Total number of outcomes: 5 (there are 5 marbles in total)

• So the probability = 4 = 0.8

Probability Line

Probability is Just a Guide

• Probability does not tell us exactly what will happen, it is just a guide

• Example: toss a coin 100 times, how many Heads will come up?

• Probability says that heads have a ½ chance, so we can expect 50 Heads.

• But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Definition

• Experiment or Trial: an action where the result is uncertain.

• Sample Space: all the possible outcomes of an experiment

• Event: a single result of an experiment

Exercise

• Which of the following is an experiment? • Tossing a coin.• Rolling a single 6-sided die.• Choosing a marble from a jar.• All of the above.

Exercise

• Which of the following is an outcome? • Rolling a pair of dice.• Landing on red.• Choosing 2 marbles from a jar.• None of the above.

• Which of the following experiments does NOT have equally likely outcomes?

• Choose a number at random from 1 to 7.

• Toss a coin.

• Choose a letter at random from the word SCHOOL.

• None of the above.

• What is the probability of choosing a vowel from the alphabet?

• A number from 1 to 11 is chosen at random. What is the probability of choosing an odd number?

Exercise

• A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?

Exercise

• A card is drawn at random from a deck of cards. Find the probability of getting

• A) the 3 of Heart• B) a Diamond• C) a Queen

Throwing dice

• If you throw 2 dice, what is the probability• A)to have a double(2 same numbers)• B) to have sum of 1• C)to have sum less than 13• D) to have sum of 4• E) to have sum less than 12• F) to have sum greater than 8

• A drawing pin can land either point down or point up.If drop it 100 times and found that it landed point up 38 times. What is the approximate probability the pin lands point up?

• What if you drop the pin two times, what is the probability

• if it land on point up for 2 times ?• If it land on point up and point down for each

time?

• The probability of raining tomorrow is 0.7. if it will rain tomorrow then the probability of raining the day after tomorrow is 0.8. Otherwise, the the probability of rain the day after tomorrow is 0.2. What is the probability of raining the day after tomorrow if today is sunny? What is the probability of it rains two days in a row?

Jack wakes up late on average 3 days in every 5 days. If Jack wakes up late the prob he is late for school is equal 9/10. If he does not wake up late, the probability he is late for school is 3/10. On what percent of days does Jack get to school on time.