Probability

Predictions

Ch. 1, Act. 5

Probability

• The study of random events.• Random events are things that happen without

predictability – e.g. the flip of a coin.• Random events in large numbers are more

predictable

Determining Probability

• Probability (P) of an event is defined as the ratio of the number of ways a desired outcome may occur divided by the total number of possible outcomes:

Number of ways to obtained desired outcomeTotal number of possible outcomes

• Probability, 0 < P < 1• Note that it cannot be greater than 1 or less than

0

P =P =

A Flip of a Coin

• What is the probability of getting a heads on any flip of the coin?

Number of ways to obtained desired outcomeTotal number of possible outcomes

1 head1 head or tails

Since 1 head + 1 tail = 2 possible outcomes.

P = ½ = 0.5

P =P =

P =P =

Roll of the Dice

• What is the probability of rolling a 5?• Since there are 6 sides to a die, and there is

only one side with a 5, the probability is:

P = 1/6

• What is the probability of rolling a 2 or a 5?• Since there are 6 sides to a die, and there are is

a side each with a 2 and a 5, the probability is:

P = 2/6, or 1/3 (0.33)

A Deck of Cards

• What is the probability of pulling an ace from a deck of cards?• Since there are 4 aces in a deck of 52 cards:

P = 4/52 = 1/13

• What is the probability of pulling an ace of spades from a deck of cards?• Since there is only one ace of spades in a deck of 52

cards:

P = 1/52

Predictability of Random Events

• While the flip of a coin, roll of a dice or a hand of poker cannot be determined from one flip, roll or hand to the next, many coin tosses, roll of the dice or hands in poker can be determined with a relatively accurate level of predictability.• What does this mean?

• As you increase the number of experimental trials, the outcome of an event becomes more predictable, and aligned with the theoretical prediction.

How can we predict multiple coin tosses?

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

Pascal’s Triangle

Activities 3 & 4 Revisited

• In Activity 3, you discovered a pattern. • If you took only one measurement, could you have

concluded that the circumference to diameter ratio was a constant?

• With our “paper toss”, would you have been as convinced of the outcome with only one run?