Statistics3.2 Conditional Probability and the Multiplication Rule
LEQ:How do you know if 2 events are independent or dependent of one another?
Procedure:
- Conditional Probability:
- Definition 1: A ______is the probability of an event occurring, given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is denoted by P(B/A) and is read as “probability of B, given A.”
- Examples 1 & 2: Finding conditional probabilities:
- Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced.)
- The table below shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ, given that the child has the gene.
Gene
present / Gene not
present / Total
High IQ / 33 / 19 / 52
Normal IQ / 39 / 11 / 50
Total / 72 / 30 / 102
- Definition 2:Two events are ______if the occurrence of one of the events does not affect the probability of the occurrence of the other event. Two events A and B are independent if
Events that are not independent are ______.
- Examples 3 – 6: Classifying events as independent or dependent:
Decide whether the events are independent or dependent.
3. Selecting a king from a standard deck (A), not replacing it, an
then selecting a queen from the deck (B).
4. Tossing a coin and getting a head (A), then rolling a six-sided die
and obtaining a 6 (B).
- Driving over 85 miles per hour (A), and then getting in a car accident (B).
- A salmon swims successfully through a dam (A) and another salmon swims successfully through the same dam (B).
- Multiplication Rule:
- Definition 3: ______: The probability that two events A and B will occur in sequence is
If events A and B are independent, then the rule can be simplified to
- Examples 7 – 11: Using the multiplication rule to find probabilities:
- Two cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen.
- A coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a six.
- The probability that a salmon swims successfully through a dam is 0.85. Find the probability that three salmon swim successfully through the dam.
- Find the probability that none of the three salmon is successful.
- Find the probability that at least one of the three salmon is successful in swimming through the dam.
- HW: p. 134 (3 – 24 mo3)