Lesson BASICS - Probability

Source code of 'BASICS - Probability'

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Introduction

Probability is a straight forward topic at school. There are only a few areas of interest at this level. First though, you need to remember some of the fundamentals:

Probability ranges from 0 (impossible) to 1 (will happen). You will NEVER get a probability beyond this range. If you do, your working is wrong somewhere.

In probability, there are 3 classic examples - a coin, a die and a pack of playing cards.

1. The Coin 
A coin has 2 possible outcomes --> HEADS or TAILS. In an unbiased coin, each result is equally likely. So,

the probability of getting heads is "1 head out of 2 possible outcomes" --> P(heads) = 1/2
the probability of getting tails is "1 tail out of 2 possible outcomes" --> P(tails) = 1/2

note: all the possible answers add up to give 1

2. The Die 
A die has 6 possible outcomes --> 1,2,3,4,5 or 6. In an unbiased die, each result is equally likely. So,

the probability of getting a 1 is "the single 1 out of 6 possible outcomes" --> P(1) = 1/6
the probability of getting a 2 is "the single 2 out of 6 possible outcomes" --> P(2) = 1/6
the probability of getting a 3 is "the single 3 out of 6 possible outcomes" --> P(3) = 1/6
the probability of getting a 4 is "the single 4 out of 6 possible outcomes" --> P(4) = 1/6
the probability of getting a 5 is "the single 5 out of 6 possible outcomes" --> P(5) = 1/6
the probability of getting a 6 is "the single 6 out of 6 possible outcomes" --> P(6) = 1/6

note: all the possible answers add up to give 1

3. The Pack of cards 
A pack of cards has 52 cards. There are a variety of patterns we could use, eg.

there are 26 red and 26 black --> half and half
there are 4 suits --> clubs, diamonds, hearts and spades
there are 13 cards in each suit
there are 4 of each number of card eg "6 of clubs", "6 of diamonds", "6 of hearts" and "6 of spades".

The probability of picking a heart at random is "there are 13 hearts out of 52 cards" OR "there are 4 suits, one of which is hearts". Both give the same answer --> P(hearts) = 1/4

EXAMPLES
Find the following probabilities, given that all items are standard and non-biased. Any card picked is also returned and the pack is shuffled too after every choice.

1. probability of a head, tossing a coin.
2. probability of rolling a 4 on a die.
3. probability of picking a red card
4. probability of picking an ace
5. probability of picking a diamond
6. probability of picking the "9 of clubs"

solutions:
1. P(head) = 1/2
2. P(4) = 1/6
3. P(red) = 26/52 --> 1/2
4. P(ace) = 4/52 --> 1/13
5. P(diamond) = 13/52 --> 1/4
6. P(9 of clubs) = 1/52

EXAMPLES
A bag contains 1 yellow, 3 red and 4 blue marbles. Picking one marble at random and then replacing it in the bag, find the following probabilities:

1. probability of picking the yellow marble.
2. probability of picking a red marble.
3. probability of picking a blue marble.
4. probability of not picking a red marble.

solutions:
1. P(yellow) = 1/8
2. P(red) = 3/8
3. P(blue) = 4/8 --> 1/2
4. P(not red)? well not red means it can be either yellow or blue. How many is that? it is 1+4 marbles... ie 5 out of 8 marbles. So, P(not red) = 5/8

The following is a very important fact to remember:
P(event) + P(not event) = 1

This means that whatever the probability is for "an event to occur", the probability of "the event not occuring" is enough to make both probabilities add to 1. This is because the event happens or it doesn't happen - together there is no other possible outcome, so together their probabilities have to add up to 1.

so. going back to question 4, P(not red) is 1-P(red)
--> P(not red) = 1 - 3/8
--> P(not red) = 5/8

which is the same answer we got by thinking about the yellow and blue marbles.

Conclusion
This is the end of the basics of Probability. If we stopped here, the topic would be pretty useless in real world situations, so the next lesson will advance the theory a little into such situations as:

"find the probabily of rolling a 2 or a 3 on a die" 
or 
"what is the probability of rolling a 2 and also picking an ace from a pack of cards".