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Basics: The chances or probability of anything Is measured on a scale of 0 to 1. A certainty is 1, an impossibility is 0. The probability of an event happening may be expressed as a fraction, ie. the probability of predicting a number on a throw of a dice is one in six and expressed as
. The sum of all probabilities is 1, ie. 6 sides to a dice, 6 probabilities, 6 x
If the probability of an event hapening is 'p', then the probability that it will not happen is 1 - p, ie. the probability that the prediction of the dice throw is wrong is 1 -
.
If two independant events have the probabilities 'p' and 'q' then the probability of both events happenning is 'p x q', ie. the probability of predicting the numbers of two consecutive dice throws are
. The general method to calculate these type of probabilities is to construct a 'Tree Diagram'.
Complementary Events: An event may take place, or not take place. As in a dice a six may be thrown, or not be thrown. The probabilities of the event and non event all add up to 1. Therefore the probability of a six thrown is:
Combined Events: Two or more events occurring together, or one after the other. In this case a coin may be tossed, heads or tails, and a dice thrown for a six. The probability outcome is made by combining the outcomes of the separate events. A separate table may show all possible outcomes.
(12 Possibilities) Heads 1 Heads 2 Heads 3 Heads 4 Heads 5 Heads 6 Tails 1 Tails 2 Tails 3 Tails 4 Tails 5 Tails 6
There are 6 possible die outcomes each having a further 2 coin outcomes, giving a combined outcome of 12 possibilities. Therefore the probability of throwing a six and 'heads' is one outcome in twelve:
. If the problem was changed to the possibility of throwing an odd number and 'tails', the table will show that this may occur 3 times, (1 tails, 3 tails, 5 tails) out of twelve outcomes, and therefore the probability is reduces to 3
12, or
.
Mutually Exclusive Events: Two or more events that cannot occur at the same time, ie. the throwing of a six cannot occur at the same time as the throwing of another number. This can be shown by probability P(1) + P(2). As seen above the probability is calculated as a fraction brought down to its lowest terms.
What is the probability that a card picked from a pack of 52 is an ace, or the king of clubs? This is a mutually exclusive event, only one card is selected. Therefore the probability: P(king of clubs) + P(ace). There is only one king of clubs therefore the outcome is:
, there are four aces therefore the outcome is:
. P(
.
Independant Events: Two or more events that do not effect each other, (the outcome of the second is not reliant on the first), ie. the throwing of a dice twice. This can be shown by probability P(1) multiplied by P(2).
What is the probability of throwing a six and then throwing a one? The two events are independant, one action does not determine the next. There are six numbers to a dice therefore both events have a one in six possible outcome. For both to occur:
What is the probability of tossing a coin and it landing on 'heads' for six consecutive events? Independant events are: P(1) x P(2) x P(3) x P(4) x P(5) x P(6), one event is one in two possible outcomes:
, therefore:
.
Tree Diagrams: Is used to solve problems with combined events.
Example: A drawer has 6 blue socks and 10 red socks, what is the probability of:
(i) getting two blue socks; (ii) getting two red socks; (iii)one sock of each colour.
Answer: As the second sock event is dependant on the first sock event and is conditional, also the events are mutually exclusive. Draw a probability tree diagram, and enter the probable outcomes.
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