Probability
Basic Probability
When you roll a die you may be interested in throwing a prime number. When you draw a name out of a hat, you may want to draw a boy`s name. Throwing a prime number or drawing a boy`s name are examples of events.
Probability is a measure of how likely an event is to happen. Something that is impossible has a value of zero and something that is certain has a value of one. The range of values from zero to one is called a probability scale. A probability cannot be negative or be greater than one.
The smaller the probability, the closer it is to zero and less likely the associated event is to happen. Similarly,the higher the probability, the more likely the event.
Performing an experiment, such as rolling a die, is called a trial. If you repeat an experiment, by carrying out a number of trials, then you can find an experimental probability of an event happening: this fraction is often called the relative frequency.
$ \displaystyle P(A)=\frac{{number~of~times~desired~event~happens}}{{number~of~trials}}$ or, sometimes
$ \displaystyle P(A)=\frac{{number~of~successes}}{{number~of~trials}}$
Example 1: Suppose that a blindfolded man is asked to throw a dart at a dartboard. If he hits the number six 15 times out of 125 throws, what is the probability of him hitting a six on his next throw?
$ \displaystyle P(six)=\frac{{number~~times~~a~~six~~obtained}}{{number~~of~~trials}}~~$
$ \displaystyle P(six)=\frac{{15}}{{125}}$
$ \displaystyle P(six)=0.12$
Relative frequency and expected occurrences
You can use relative frequency to make predictions about what might happen in the future or how often an event might occur in a larger sample. For example, if you know that the relative frequency of rolling a 4 on particular die is 18%, you can work out how many times you would expect to get 4 when you roll the dice 80 or 200 times.
18% of 80=14.4 and 18% of 200=36, so if you rolled the same die 80 times you could expect to get a 4 about 14 times and if you rolled it 200 times, you could expect to get a 4 thirty-six times.
Remember through, that even if you expected to get a 4 thirty six times, this is not given and your actual results may be different.
Theoretical probability
When you flip a coin you may be interested in the event obtaining a head but this is only one possibility. When you flip a coin there are two possible outcomes, obtaining a head or obtaining a tail.
You can calculate the theoretical (or expected) probability easily if all of the possible outcomes are equally likely, by counting the number of favorable outcomes and dividing by the number of possible outcomes. Favorable outcomes are any outcomes that mean your event has happened.
For example, if you throw an unbiased die and need the probability of an even number then the favorable outcomes are two, four or six. There are three of them.
Under these circumstances the event A (obtaining an even number) has the probability:
$ \displaystyle P(A)=\frac{{number~of~favourable~outcomes}}{{number~of~possible~~outcomes}}$
Of course a die may be weighted in some way, or imperfectly made, and indeed this may be true of any object discussed in probability question. Under these circumstances a die, coin or other object is said to be biased. The outcomes will no longer be equally likely and you may need to use experimental probability probability.
Example 2: An unbiased die is thrown and the number on the upward face is recorded. Find the probability of obtaining:
a) a three
There is only way of throwing a three, but six possible outcomes (you could roll a 1, 2, 3, 4, 5, 6).
$ \displaystyle P(3)=\frac{1}{6}$
b) an even number
There are three even numbers on a die, giving three favourable outcomes.
$ \displaystyle P(even~~number)=\frac{3}{6}=\frac{1}{2}$
c) a prime number
The prime numbers on a die are 2, 3 and 5, giving three favourable outcomes.
Example 3: A card is drawn from an ordinary 52 card pack. What is the probability that the card will be a king?
Number of possible outcomes is 52. Number of favorable outcomes is four, because there are four kings per pack.
$ \displaystyle P(King)=\frac{4}{{25}}=\frac{1}{{13}}$
Example 4: Jason has 20 socks in a drawer. 8 socks are red, 10 socks are blue and 2 socks are green. If a sock is drawn at random, what is the probability that it is green?
Number of possible outcomes is 20.
Number of favorable outcomes is 2.
$ \displaystyle P(green)=\frac{2}{{20}}=\frac{1}{{10}}$
Example 5: Nine painters are assigned a letter from the word HOLLYWOOD for painting at random. Find the probability that a painter is assigned:
For each of these the number of possible outcomes is 9.
a) the letter `Y`
Number of favourable outcomes is one (there is only one `Y`)
$ \displaystyle P(Y)=\frac{1}{9}$
b) the letter `O`.
Number of favourable outcomes is three.
$ \displaystyle P(O)=\frac{3}{9}=\frac{1}{3}$
c) the letter `H` or the letter `L`.
Number of favourable outcomes=number of letters that are either H or L=3, since there is one H and two L`s in Hollywood.
$ \displaystyle P(H~or~L)=\frac{3}{9}=\frac{1}{9}$
d) the letter `Z`.
Number of favorable outcomes is zero (there are no `Z`s)
$ \displaystyle P(H~or~L)=\frac{3}{9}=\frac{1}{9}$
The probability that an event does not happen
Something may happen or it may not happen. The probability of an event happening may be different from the probability of the event not happening but the two combined probabilities will always sum up to one.
If A is an event, then $ \displaystyle \overline{A}$ is the event that A does not happen and $ \displaystyle P\left( {\overline{A}} \right)=1-P\left( A \right)$.
Example 6: The probability that James passes his driving test is $ \displaystyle \frac{2}{3}$. What is the probability that James fails?
P(failure)=P(not passing)=1-P(passing)
$ \displaystyle P(failure)=1-\frac{2}{3}=\frac{1}{3}$