Probability INDEPENDENT EVENTS

Independent Events  Life is full of random events!  You need to get a "feel" for them to be a smart and successful person.  The toss of a coin, throwing dice and lottery draws are all examples of random events.

Dependent Events  Sometimes an event can affect the next event.  Example: taking colored marbles from a bag: as you take each marble there are less marbles left in the bag, so the probabilities change.  We call those Dependent Events, because what happens depends on what happened before

Independent Events  Independent Events are not affected by previous events This is an important idea!! A coin does not ‘know’ that it came up heads before Each toss of a coin is a perfectly isolated thing

Independent Events  Example: You toss a coin and it comes up "Heads" three times... what is the chance that the next toss will also be a "Head"?  The chance is simply ½ (or 0.5) just like ANY toss of the coin.  What it did in the past will not affect the current toss!

Independent Events  Some people think "it is overdue for a Tail", but really truly the next toss of the coin is totally independent of any previous tosses.  Saying "a Tail is due", or "just one more go, my luck is due" is called The Gambler's Fallacy

Recall  Probability = Number of ways it can happen Total number of outcomes

Example  What is the probability of getting a head when tossing a coin?  Number of ways it can happen (1 – Head)  Total number of possible outcomes (2 – Head or Tail)  So the probability is ½ or 0.5

Example  What is the probability of getting a ‘5’ or a ‘6’ when rolling a die?  Number of ways it can happen 2 - (5 and 6)  Total number of outcomes 6 – (1, 2, 3, 4, 5, 6)  So the probability is 2/6 or 1/3

Writing Independent Probabilities  As we have already seen, probabilities exist between 0 and 1  As such, we can represent probabilities as decimals or fractions  We can also represent probabilities as percentages

Two or More Independent Events  What happens when two or more independent events are happening?  For example, what is the probability of tossing a coin 3 times and getting three tails as a result?

How can we show this?  Using a Tree Diagram!  A tree diagram is a diagram used to help figure out the probability when there is more than one events happening  Let’s make a tree diagram for a coin toss

Tree Diagram T H THTHTHTH THTHTHTHTHTHTHTH Toss 1 Toss 2 Toss 3 Possibilitie s T – T – T T – T – H T – H – T T – H – H H – T – T H – T – H H – H – T H – H – H Coin FlipFrequency Three Tails1 Three Heads1 Two Heads, One Tail 3 Two Tails, One Head 3 Total Flips8 P(Three Tails) = Frequency / Total P(Three Tails) = 1/8

Example  Your friend invites you to a movie, saying it starts sometime between 4pm and midnight on Saturday or Sunday. What are the chances that it starts on Saturday between 6 and 8?  Let’s make a tree diagram

Saturday Sunday P(Saturday 6 – 8) = Number of Saturday 6 – 8 Total Possibilities P(S 6-8) = 1 8

Example  A coin is tossed three times  Find the probability of getting at least 2 heads

Example: A coin is tossed three times Find the probability of getting at least 2 heads THTH THTHTHTH THTHTHTHTHTHTHTH Toss 1 Toss 2 Toss 3 Possibilitie s T – T – T T – T – H T – H – T T – H – H H – T – T H – T – H H – H – T H – H – H Coin FlipFrequency Three Tails1 Three Heads1 Two Heads, One Tail 3 Two Tails, One Head 3 Total Flips 8 P(At least two H) = Frequency / Total P(At least two H) = 4/8 = ½

Two or More Independent Events  We can calculate the chances of two or more independent events by multiplying their individual probabilities  What is the Probability of flipping a coin Tails three times?  P(Tails) = ½  P(Tails 3 Times) = ½ x ½ x ½ = 1/8 or 12.5%  Although the individual was ½, the probability for it happening three times in a row significantly decreased

Two or More Independent Events  Thus, we can calculate the probability of multiple independent events by multiplying  P(A and B) = P(A) x P(B)  For the movie example, P(Saturday) = ½ P (Your Time 6-8) = 2/8 P(Saturday and Your Time) = ½ x 2/8 = 1/8

Example  Two cards are drawn from the top of a well-shuffled deck. What is the probability that they are both black aces?

Example  A die is thrown twice. What is the probability that both numbers are prime?

Example  A code consists of a digit from 0-9 followed by a letter. What is the probability that the code is 9Z?

Example  A bag contains 5 red marbles, 4 green marbles and 1 blue marble. A marble is chosen at random from the bag and not replaced; then a second marble is chosen. What is the probability both marbles are green?

Multiplication Law for Independent Events  When finding probabilities for multiple independent events, we use multiplication to find them  For example, when we are looking for the probability of this event AND this event to happen, we multiply their individual probabilities

Independent Events  There is also the possibility that we are looking for the probability of this event OR this event to happen  When this is the case, we add the probabilities  For example, A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that: a) Both marbles are the same colour b) As least one of the marbles is green c) The marbles are different colours

Example  A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that: a) Both marbles are the same colour b) As least one of the marbles is green c) The marbles are different colours P(RR or GG)

Example  A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that: a) Both marbles are the same colour b) As least one of the marbles is green c) The marbles are different colours P(RG or GG or GR)

Example  A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that: a) Both marbles are the same colour b) As least one of the marbles is green c) The marbles are different colours Note: We are looking for the probability of P(RG or GR or GG). In this example, there are only four different possible outcomes. We also know that the probabilities must add to be 1. So we can also find the probability by doing 1 – P(RR)  Using subtraction to find probability can be easier than finding direct probabilities sometimes

Example  A bag contains 4 red marbles and 3 green marbles. One marble is drawn at random and then put back. A second marble is drawn. What is the probability that: a) Both marbles are the same colour b) As least one of the marbles is green c) The marbles are different colours P(RG or GR)

Example  Sylvia drives through three sets of lights on her way to work. The probability of each set being green is 0.3. What is the probability that all three sets are green?