Download Probability Probability is

 Probability Probability is the likelihood (chance) of an event happening.
Probability is measured on the probability scale
probability scale
Impossible
0
Even
Chance
1
2
Certain
1
1
At its most basic we can say whether events are possible or impossible, likely or unlikely.
e.g The probability that the prime minister will walk down to Bournemouth beach tomorrow is unlikely,
where as the probability the prime minister will walk through the front door of number 10 downing street tomorrow is likely.
The probability that a pig will fly over london tomorrow is impossible
where as the probability that a bird will fly over london tomorrow is certain.
Impossible, even chance and certain are all exact measures of probability. Other words that we associate with probability like likely, unlikely, possibly, maybe....are not exact measures of probability.
All of these words in red make up the spaces in between the blue words on the probability scale that was on the previous page.
We need to be more precise when it comes to probability statements. 2
Calculating probabilities To find exact probabilities that lie in the spaces between the 3 key phrases on the probability scale we can use the following:
write the probability of events happening as a FRACTION.
Probability = Number of desired outcomes
Total number of possible outcomes
e.g 1 If I roll a dice, what is the probability that it lands on a 5?
There is ONE desired outcome ­ landing on a 5
There are SIX possible outcomes ­ 1,2,3,4,5,6
So using the above formula:
Probability of rolling a 5 = 1
6
3
e.g 2 What is the probability of rolling a number greater than 2? There are four possible outcomes greater than 2 on the dice: 3, 4, 5 and 6
As before there are still 6 possible outcomes in total
This means that the Probability = 4
6 e.g 3 What is the probability of rolling the dice and landing on a number that is NOT a 1? There are 5 outcomes which are NOT a 1: 2, 3, 4, 5 and 6
This gives us the probability 5
6
The probability of an event NOT happening = 1 ­ probability of the event happening
We can see that the formula in red is true from the above example as:
6 ­ 1 = 5
6 6 6
1 = 66
remember: 4
e.g 2 If we spin the following spinner, what is the probability of the arrow pointing at a number 2?
1
3
1
2
2
1
3
With this example we need to be careful as there are only 7 sections indicated inside the spinner but they are NOT all the same size
You need to draw in this dotted line to see that the number 2 section on the original is actually TWO sections. Now we have 8 sections which are the SAME SIZE.
1
3
2
1
2
2
1
3
Now we use the formula as before.
There are 3 desired outcomes (3 sections numbered 2)
There are 8 sections in total Probability of landing on a 2 = 3
8
5
If there is a situation where there is a finite number of possibilities then all of the probabilities of all the possibilities must add up to 1
The reason for this is because of there is only a set number of outcomes and you are going to apply one of them then at least one of them is certain to happen.
We can apply this knowledge to solve problems involving probability.
e.g There are only red, yellow and green counters in a bag. I am going to pick one of them at random. The probability of picking a red counter is 1
7
Yellow is twice as likely to be picked than green
What is the probability of picking a green counter?
We know that red = 1 which means the probability of not a red = 6
7 7
Not a red means picking a yellow or green.
If Yellow is twice as likely as green then we need to split 6 into the ratio 2:1
7
Revise ratio to help with this
This gives us: 4 : 2
7 7
So the probability of getting a green is 2
7
2 + 4 + 1 = 7
7 7 7 7
green + yellow + red = certain
6
Mutually exclusive events Jim and Bob are playing a game. They will roll a dice once. If the dice lands on an even number then bob will win a prize. If the dice lands on an odd number then Jim will win a prize.
Is it possible for BOTH to win a prize at the same time?
Bob needs a 2, 4 or 6 to be rolled
Jim needs a 1, 3 or 5 to be rolled
There is no number that exists which satisfies both outcomes. This means that these events ARE mutually exclusive.
e.g 2 If I have 10 cards in a bag numbered 1 to 10 and I pick one at random, is it possible to pick out a card at random that is an even number AND a prime number?
Even numbers ­ 2, 4, 6, 8, 10
Prime numbers ­ 2, 3, 5, 7
YES it is, as 2 is BOTH even and prime
This means that it is possible for these results to overlap so these events are NOT mutually exclusive.
7
Experimental and Theoretical Probability In THEORY if I roll a dice the probability of getting a number is one in six. So in THEORY if I roll a dice six times each number should come up once. However, if I was to carry out this EXPERIMENT then there is a good chance this will not happen.
However, the more times you carry out an experiment the closer the number of outcomes will get to the theoretical probability.
e.g I spin the following spinner 100 times.
It lands on the following:
Blue ­ 23 times
Red ­ 35 times
Yellow ­ 17 times
Green ­ 25 times
Red
Yellow
Blue
Green
Based on this experimental evidence if I was asked what is the probability of this spinner landing on red I would have to say 35
100
However theory says, correctly, that the probability is a quarter.
If I was to spin the spinner 10000 times I would get results that are much closer to the theoretical probability
8
Probability of successive events To calaulate the probability of events happening in a certain order (successive events) we need to work out the probability of each event happening individually and then MULTIPLY these probabilities together.
e.g If I roll a dice 3 times, what is the probability of rolling a 1, followed by an even number, followed by a prime number?
We need to calculate each probability seperately first:
Prob of rolling a 1 = 1
6
Prob of rolling an even number = 1
2
Prob of rolling a prime number = 1
2
We now multiply these probabilities together:
You might need to revise Multiplying fractions
to be able to do this
1 x 1 x 1 = 1
6 2 2 24
Answer = 1 24
9
Probability trees We can use a probability tree to calculate the probability of 2 successive events.
In order to do this we multiply the probabilities together.
e.g If I roll the same dice twice in a row, what is the probability of getting an odd number followed by the number 5?
P(of a 5)
P(an odd)
1
2
1
6
P(not a 5) 5
1 6
P(of a 5) 6
P(not an odd)
1
2
P(not a 5)
5
6
We need to multiply the probabilities together
1 x 1 = 1
2 6 12
10