Probability Concepts on CFA Level 1 can sometimes pose challenges to some candidates. Statistics and related concepts will appear throughout the three levels. It is important that you understand the probability concepts on CFA Level 1 in order to have a good foundation for the two other exams. In this article we will review the most important probability concepts related to Probability of a Single Event, Probability of Two (or more) Independent Events, Conditional Probabilities, Tree Diagrams and the Bayes’ Formula.
Important Terms Related To Probability Concepts On CFA Level 1
- Random variable: uncertain quantity/number
- Outcome: observed value of a random variable
- Event: single outcome or set of outcomes
- Mutually exclusive events: events that cannot occur at the same time
- Exhaustive events: list of events that includes all possible outcomes
Probability of a Single Event
If you roll a six-sided die, there are six possible outcomes, and each of these outcomes is equally likely. A six is as likely to come up as a three, and likewise for the other four sides of the die. What, then, is the probability that a one will come up?
Since there are six possible outcomes, the probability is 1/6. What is the probability that either a one or a six will come up? Given that all outcomes are equally likely, we can compute the probability of a one or a six using the formula:
In this case there are two favorable outcomes and six possible outcomes. So the probability of throw- ing either a one or six is 1/3. The above formula has many applications in the investment field and can also be applied to many games of chance.
For example, what is the probability that a card drawn at random from a deck of playing cards will be an ace? Since the deck has four aces, there are four favorable outcomes; since the deck has 52 cards, there are 52 possible outcomes. The probability is therefore 4/52 = 1/13.
If you know the probability of an event occurring, it is easy to compute the probability that the event does not occur. If P(A) is the probability of Event A, then 1 – P(A) is the probability that the event does not occur.
Probability of Two (or more) Independent Events
Events A and B are independent events if the probability of Event B occurring is the same whether or not event A occurs.
Let’s take a simple example. A fair coin is tossed two times. The probability that a head comes up on the second toss is 1/2 regardless of whether or not a head came up on the first toss. The two events are (1) first toss is a head and (2) second toss is a head. So these events are independent.
Probability of A and B
When two events are independent, the probability of both occurring is the product of the probabilities of the individual events. More formally, if events A and B are independent, then the probability of both A and B occurring is:
P(A and B) = P(A) × P(B)
- P(A and B) is the probability of events A and B both occurring
- P(A) is the probability of event A occurring
- P(B) is the probability of event B occurring
If you flip a coin twice, what is the probability that it will come up heads both times? Event A is that the coin comes up heads on the first flip and Event B is that the coin comes up heads on the second flip. Since both P(A) and P(B) equal 1/2, the probability that both events occur is:
1/2 × 1/2 = 1/4
Probability of A or B
If Events A and B are independent, the probability that either Event A or Event B occurs is:
P(A or B) = P(A) + P(B) – P(A and B)
In this discussion, when we say “A or B occurs” we include three possibilities:
- A occurs and B does not occur
- B occurs and A does not occur
- Both A and B occur
For example, if you flip a coin two times, what is the probability that you will get a head on the first flip or a head on the second flip (or both)? Letting Event A be a head on the first flip and Event B be a head on the second flip, then P(A) = 1/2, P(B) = 1/2, and P(A and B) = 1/4. Therefore,
P(A or B) = 1/2 + 1/2 – 1/4 = 3/4.
If you throw a six-sided die and then flip a coin, what is the probability that you will get either a 6 on the die or a head on the coin flip (or both)? Using the formula,
P(6 or head) = P(6) + P(head) – P(6 and head)
= (1/6) + (1/2) – (1/6)(1/2)
Often it is required to compute the probability of an event given that another event has occurred. For example, what is the probability that two cards drawn at random from a deck of playing cards will both be aces? It might seem that you could use the formula for the probability of two independent events and simply multiply 4/52 x 4/52 = 1/169. This would be incorrect, however, because the two events are not independent. If the first card drawn is an ace, then the probability that the second card is also an ace would be lower because there would only be three aces left in the deck.
Once the first card chosen is an ace, the probability that the second card chosen is also an ace is called the conditional probability of drawing an ace. In this case, the “condition” is that the first card is an ace. Symbolically, we write this as:
P(ace on second draw | an ace on the first draw)
The vertical bar “|” is read as “given,” so the above expression is short for: “The probability that an ace is drawn on the second draw given that an ace was drawn on the first draw.” What is this probability? Since after an ace is drawn on the first draw, there are 3 aces out of 51 total cards left. This means that the probability that one of these aces will be drawn is 3/51 = 1/17.
If Events A and B are not independent, then:
P(A and B) = P(A) x P(B|A)
Applying this to the problem of two aces, the probability of drawing two aces from a deck is:
4/52 x 3/51 = 1/221
Tree diagrams can be used to represent an investment problem with various probabilities of outcome:
This tree diagram illustrates the earnings per share of a firm given various economic scenarios. The total probabilities of outcomes should add up to 100% and the expected EPS is given by computing the weighted average of all the end scenarios’ EPS.
This formula is used to update probability given the addition of new information. The result is the up- dated probability of an event and is expressed as:
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