# A Comprehensive Guide to Calculations on Probability

Probability simply means ‘chance’ or  ‘likelihood.’ For example, if you toss a coin, it is likely to show either a head (H) up or tail (T) up but you cannot tell before hand that it will show head up or tail up for sure.

In this tutorial, you will learn how to solve questions on the different types of probability including experimental probability, theoretical probability, mutually exclusive events and independent events.

BONUS: A link to real past questions on probability is included at the end of this tutorial.

## BASIC TERMS USED IN PROBABILITY

• A RANDOM EXPERIMENT: Any experiment whose outcome cannot be predicted in advance is a random experiment. For example, tossing of a coin
• A TRIAL: Each performance in a random experiment is called a trial. For example, each toss of a coin is a trial.
• OUTCOME: This is the result of a trial in a random experiment. For example, what you get (head or tail) from tossing a coin is the outcome.
• SAMPLE SPACE: This is the set of all the possible outcomes of random experiment. For example, when you toss a coin, there are ONLY two possible outcomes to expect which are either a head (H) or a tail (T). These outcomes are the sample space of the experiment of tossing a coin.

## 1. Experimental Probability

Experimental probability bases its result on the actual trials carried out. Therefore, the outcome of experimental probability depends on the number of trials carried out.

FORMULA:

\dfrac{required outcome}{possible outcome}

QUESTION:

Out of 500 fruits bought in a market, 25 are bad. What is the probability that a fruit picked from the lot will be good?

SOLUTION:

Since the total is 500, and out of this total, 25 are declared bad.

Hence, 500 – 25 = 475

The probability of picking good fruits is = \dfrac{475}{500} which becomes \dfrac{19}{20} when reduced to its lowest form.

## 2. Theoretical Probability

Theoretical probability bases its results on the exact values that are dependent on the physical nature of the conditions or situations that are being considered.

Supposing you toss a coin and roll a die together, the two possible outcomes for your coin toss will be either a Head (H) or a Tail (T). And for the die, the possible outcomes will be either 1,2,3,4,5, or 6

QUESTION:

Three coins are thrown together, find the probability of getting two head and one tail.

Remember the coins are three, so your possible outcomes from tossing these coins will be:

1. HHH
2. HHT
3. HTH
4. THH
5. HTT
6. THT
7. TTH
8. TTT

These are the 8 possible outcomes that you will get from the three coins altogether.

SOLUTION:

Therefore, the probability of getting two heads and a tail will be equal to the number of places with 2 heads and 1 tail/total number of outcomes = \dfrac{3}{8}

## 3. Mutually Exclusive Event

Two event, X and Y are said to be mutually exclusively if the occurrence of X prevents the occurrence of Y.

In questions that involve mutually exclusive events, word such as “Or” “either” “neither” are used.

Mutually exclusive events are otherwise called ADDITION OF PROBABILITY because when the occurrence of two or more event are required, the separate probabilities of such events are first found and then added together.

QUESTION:

A bag contains 5 red balls, 6 yellow balls, and 4 white balls. A ball is picked from the bag at random. Find the probability that the ball is Red or White.

SOLUTION:

Total number of balls = 5 + 6 + 4 = 15 balls

P(red or white) = P(red) + P(white) = \dfrac{5}{15} + \dfrac{4}{15} = \dfrac{9}{15} which becomes \dfrac{3}{5} when reduced to its lowest form.

## 4. Independent Events

If two events can occur without affecting each other, then the two events are said to be independent events.

Independent event in probability is otherwise called multiplication of probability. In questions involving independent events, word such as “and” “both” “all” are usually used.

For example, can you clap your hands and sing at the same time?

Yes, it is possible to do each of the two events above at the same time without one affecting the other.

QUESTION:

Two dice are thrown once, what is the probability of getting 2 six.

SOLUTION: Note that the two dice are two separate items having their own probability of each number on it to be equal to \dfrac{1}{6}.

The probability of getting 2 six is equal to the probability of 6 on the first die = \dfrac{1}{6} and the probability of 6 on the second die = \dfrac{1}{6}

Therefore, the probability of 6 on the two dice = \dfrac{1}{6} × \dfrac{1}{6} = \dfrac{1}{36}

## Summary of probability

1. The probability of any event is greater than or equal to zero.
2. The sum of probability of two or more events is equal to 1
3. The probability of certainty is 1
4. The probability of uncertainty is 0
5. If the probability that an event will certainly occur is P, then the probability that the event will not occur is donated by q = 1 – P

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