Grade 12 – Data Management
Experimental and Theoretical Probability
- Probability: is the value between 0 and 1 that describes that likelihood of an occurrence of a certain event.
- Experimental Probability: making predictions based on a large number of previous results.
- Theoretical Probability: Make predictions based on a mathematical model.
- In general, experimental probability will approach theoretical probability as the number of trials increase.
- Discrete Sample Space: a sample space where you can count the number of outcomes ie. blue balls
- Continuous Sample Space: decimal numbers with infinite possibilities ie. Time.
- Event: is the occurrence of a specific outcome in the sample space.
P(A) = n(A) / n(S)
Probability of A is number of outcomes for A over total possibilities
- P(A’) the probability that event A will not occur.
- P(A’) = 1 – P(A)
- Odds: a ratio used to represent a degree of confidence in whether or not an event will occur.
- Odds In favour: P(A) : P(A’)
- = n(A) : n(A’)
- Odds Against: P(A’) : P(A)
- = n(A’) : n(A)
Probability using counting principles
- Instead of listing out all possibilities, counting principles such as combinations and permutations can be used to calculate all the possibilities of outcome and the possibilities of the event occurrence.
- Refer to these links for information about counting principles
Independent and Dependent Event
- Two events are independent if the occurrence of one event has no effect on the occurrence of another event.
- If two events are independent, then P (A n B) = P(A) P(B)
- Drawing tree diagrams with probability percentages on the branches can be multiplied
- P(AA) = P(A)*P(A)
- ie. When drawing disks from a bag, if the disks are replaced, the 2nd draw will be an independent event.
- ie. When drawing disks from a bag, but the disks are not replaced, the 2nd draw will be a dependent event.
Mutually Exclusive Events
- Two events are mutually exclusive if when one event occurs, the other event cannot occur.
- If two events are mutually exclusive, then P(A U B) = P(A) + P(B)
- If two events are not mutually exclusive, then P( A U B) = P(A) + P(B) – P(A U B)
- ie Probability of picking a KING or a FOUR is a mutually exclusive event.
- ie Probability of picking a KING or a RED card is non-mutually exclusive.
- The probability that an event will occur given that another compatible event that already occurred.
- P(A / B) = P(A and B) / P(B)
- Probability of A given the occurrence of B is equal to the probability of A and B over the probability that B has occurred.
- ie. Probability of drawing a QUEEN if we know the chosen card is a face card is an example of conditional probability.