Theoretical Probability of Simple Events

Theoretical probability is worked out by counting rather than by experimenting. First list the sample space — every outcome that could happen. For a six-sided die that is 1 through 6. Then the probability of an event is the number of favourable outcomes over the total number of outcomes. So the probability of rolling an even number is 3 over 6, which is one half, because three of the six faces are even. One condition matters: this only works when every outcome is equally likely. A fair die, a fair coin or a well-shuffled deck qualify. 'Will it rain tomorrow' does not, because you cannot count equally likely outcomes.

Every probability sits between 0 and 1. Zero means the event cannot happen, like rolling a 7 on an ordinary die. One means it must happen, like rolling a number under 7. Everything else falls in between, with a half being an even chance. You can write a probability three ways — as a fraction like one half, a decimal like 0.5, or a percentage like 50 percent. Fractions are usually safest because nothing gets rounded, and you should simplify them, so three sixths becomes one half. The scale gives you a free check: if your answer is above 1 or negative, you have made an error.

The same formula handles much bigger sample spaces. A standard deck has 52 cards in four suits of 13, with hearts and diamonds red and clubs and spades black. The probability of drawing a heart is 13 over 52, which simplifies to one quarter. A red card is 26 over 52, or one half. An ace is 4 over 52, which is one thirteenth, because there is one ace in each suit. And the ace of spades specifically is just 1 over 52. Notice the pattern: count how many cards fit the description, then put that over 52. And always simplify — 13 over 52 is correct, but one quarter is the finished answer.

Marble questions test careful counting. A bag holds 5 red, 3 blue and 2 green, so the total is 10 — and you have to work that total out, because the question does not state it. The probability of red is 5 over 10, which is one half. Blue is 3 over 10. 'Not red' means the 3 blue plus the 2 green, so 5 over 10, again one half. 'Red or blue' is 5 plus 3, so 8 over 10, which simplifies to four fifths. The trap here is using 5 as the denominator because there are 5 red marbles. The denominator is always the total number of things in the bag — 10, not 5.

To finish, here is the routine for a full-mark answer. Work out the total number of outcomes and write it down rather than holding it in your head. Count the favourable outcomes, reading the wording carefully for words like 'or', 'not' and 'at least' which change what counts. Write the fraction with favourable on top. Simplify it. Then check it sits between 0 and 1. One more habit: use the P-bracket notation, writing 'P of red equals one half' rather than just 'one half'. It shows which event you found, which matters when a question has several parts asking about different events.