What is Probability?
In layman’s terms, probability tells us about the certainty of any particular event. For example, you can say “what is the probability that Sam will pass?" or “what is the probability that rain will not wash out the T20 final?”
Theoretically, probability of an event A is the ratio of number of favorable events (n) to number of total possible events (N). Probability of an event A can be expressed as:
P (A) = n/N
Sam can either pass or fail. The total number of possible events is 2 (N=2). And the number of favorable events (i.e. pass) is 1 (n=1). Hence, the probability for passing in case of Sam is:
1 / 2 =0.5 or 50%
- Probability of a single event: The probability of a single event A occurring is expressed as:
P (A) = Number of favorable events (n) / Total number of possible events (N)
- Probability of an event not occurring: The probability of an event not occurring is expressed as:
P (not A) = 1 – P (A)
- Probability of two independent events occurring : The probability of two events A and B, which are independent of each other, occurring can be expressed as:
P (A ∩ B) = P (A). P (B)
If two coins are tossed by two people together, then the probability of getting heads for both the coins (P (A ∩ B)) will be the multiplication of the probability of getting heads for the first coin ( P(A)) and the probability of getting heads for second coin ( P(B)). For this case joint probability will be:
P (A ∩ B) = (1 / 2)*(1/2) = 0.25
- Probability of two mutually exclusive events occurring: If any one of two events can occur at a point in time and if we are talking about calculating probability of the two events then the probability will be called the probability of two mutually exclusive events. Such probability can be expressed as:
P (A U B) = P (A) + P (B)
For example, the probability of getting either a 2 or a 6 from a six sided die can be expressed as:
P (2 or 6) = P (2 U 6) = 1 / 6 + 1 / 6 = 1/3
- Probability of two non-mutually exclusive events: If two events which are non-mutually exclusive i.e. either of the two or both of them can occur at a time, then the probability of such events can be expressed as:
P (A or B) = P (A) +P (B) – P (A and B)
For example, the probability of getting either a face or a heart or both by picking a card from the deck of cards will be:
P (Face or Heart or Both) = 12/52 +13/52 – 3/52 = 11/26
A deck of cards has 12 face cards, 13 hearts and 3 cards of both.
Procedure for Teaching Probability
- Explain the concept and the theory of probability.
- Explain the examples of dies and cards to make the subject clear.
Give the following examples for solving in the class:
- Find the probability of getting a king if a single card is being randomly chosen from a deck of 52 cards.
- What is the probability of a 30-day month in a year?
- What is the probability of rolling a four with one die? Two dice?
Blend Theory with Practical Application
Probability lessons should have the perfect blend of theory and practical examples. Most of the probability problems fall under the four categories explained in this probability lesson plan. Students need to be clear about them in order to apply them for complex problems.