Probability

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Probability
How to calculate simple probabilities
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The probability of an event happening is given by
the rule

• Probability Required Event / Total Possible
• i.e. The number of required events divided by the
  total number of possible events

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Getting a 4 on a single die

• There is 1 required event a four
• There are 6 possible events
• P(4) 1/6

The answer may be written as a fraction, decimal
or percentage
The P says, The probability of
4
The probability of not getting a 4

• There are five numbers on the single die that are
  not a 4
• There are six possible outcomes
• So, P(Not getting a 4) 5/6

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It is important that you notice that the two
probabilities add together to give 1

• The total probabilities of any event have a total
  of 1
• 1/6 5/6 1

Probability of not getting a 4 on a single die
Probability of getting a 4 on a single die
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The total of all probabilities for an event
equal 1. This important fact can be used to find
probabilities.

• For example, you are told that the probability of
  a football teams results in the next match are
• P(Win) 0.3
• P(Lose) 0.5
• What is the probability of a draw?

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Remember the total of all the probabilities is 1

• The team must win, lose or draw.
• P(Win) P(Lose) P(Draw) 1
• 0.3 0.5 P(Draw) 1

This is 0.5
This is 0.3
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To calculate P(Draw)

• P(Win) P(Lose) P(Draw) 1
• 0.3 0.5 P(Draw) 1
• P(Draw) 1 (0.3 0.5)
• P(Draw) 1 0.8
• P(Draw) 0.2

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What did you do?

• Add together the probabilities that you know and
  take this number away from one.
• You know P(Win) 0.3 and P(Lose) 0.5
• You know that the total of all the probabilities
  is 1
• So add 0.3 and 0.5 to get 0.8
• Then take this away from 1
• 1 0.8 0.2

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P(Draw) 0.2
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Can I have a date?
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Three boys ring Jane and ask her for a date
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What is the probability that she will choose Sam?
Total Probabilities 1 P(Dave) P(Glyn)
P(Sam) 1 0.1 0.6 P(Sam) 1 So P(Sam) 1
(0.1 0.6) P(Sam) 0.3
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Mutually Exclusive
Sounds very clever. It means that one event
excludes (makes impossible) another event. You
cannot get heads and tails at the same time on a
coin. You cannot win and lose at the same time.
It allows the fact that total probability is one
to be used to solve problems
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Probability Expected number of successes p
(success) number of trials Example If a die
is rolled 300 times, how many times would you
expect to roll a number greater than 4?P
(numbergt 4) 2/6 1/3Expected number of
successes p (success) number of
trialsExpected number of scores greater than 4
1/3 300 100 1230?
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 Exercise 11. A die is rolled 510 times how
many times would you expect to rolla) an even
numberb) a 4c) a number less than 4? 2. About
1/6 of people have red hair. How many red haired
people would you expect to find in a school of
1230?
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 3. The probability that Cardiff City F.C win a
match is 3/8. If they play 24 matches, how many
would you expect them to win? 4. One in seven
sweets in a bag are strawberry flavoured. If
there are 490 bags of sweets, how many times
would you expect the first sweet chosen to be
strawberry flavoured? 
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5. A pack of playing cards are spilt into two
piles. Pile A has all the picture cards and the
aces. All the other cards are in Pile B. a) If
you pick a card at random from pile A what is the
probability that it is the ace of diamonds?b)
How many times would you expect to pick a picture
card first from Pile A if you picked a card at
random 400 times and replaced the card each
time?c) How many times would you expect to pick
a heart first from Pile B, if you picked a card
at random 240 times? 6. If you remove all the
picture cards and the hearts from a deck of cards
you are left with a pile of cards. If you pick
one card at randoma)     What is the
probability that you choose the two of
spades?b)    What is the probability that the
card is a club?c)     If you were to repeat this
390 times, how many times would you expect to
pick a red card?
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7. About one in twelve people have allergies to
cats. If you asked 900 people in the street, how
many would you expect to suffer from this
allergy?  8. Thomas forgets his homework
diary one day a week (Monday-Friday). Over a
period of 30 weeks in school, how many days in
total would you expect Thomas to forget his
homework diary?