Lesson Notes By Weeks and Term v4 - SHS 3
MAKING PREDICTIONS WITH DATA
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MAKING PREDICTIONS WITH DATA
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Subject: Additional Mathematics
Class: SHS 3
Term: 2nd Term
Week: 12
Grade code: 3.4.2.LI.3
Strand code: 4
Sub-strand code: 2
Content standard code: 3.4.2.CS.1
Indicator code: 3.4.2.LI.3
Theme: HANDLING DATA
Subtheme: MAKING PREDICTIONS WITH DATA
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This lesson introduces conditional probability, a powerful tool for making predictions when we have some prior information. In our daily lives, the likelihood of something happening often changes based on another event that has already occurred. For example, the chance of traffic on the Kaneshie-Odorkor road is much higher *if* it is raining heavily. The probability of the Black Stars winning a match changes *given* that their star player is injured. Conditional probability helps us to quantify these "what if" scenarios and make more informed decisions.
a) Independent vs. Dependent Events Before we dive into conditional probability, we must understand the relationship between events. Independent Events: Two events are independent if the outcome of one event does not affect the outcome of the other. Example: Tossing a coin and rolling a die. The result of the coin toss (Heads or Tails) has absolutely no impact on the number you get on the die (1 to 6). Dependent Events: Two events are dependent if the outcome of the first event *changes* the probability of the second event. This is where conditional probability becomes essential. Example: Imagine a box containing 5 pieces of fried yam and 3 pieces of fried fish. You pick one item and eat it. Then you pick a second item. The probability of your first pick being fish is 3/8. *If* your first pick was fish, there are now only 2 pieces of fish and 7 items left. The probability of the second pick being fish is now 2/7. *If* your first pick was yam, there are still 3 pieces of fish and 7 items left. The probability of the second pick being fish is now 3/7. Clearly, the probability of the second event depends on the outcome of the first. This is a dependent event. b) What is Conditional Probability? Conditional probability is the likelihood of an event (A) occurring, *given that* another event (B) has already happened.
We write this as P(A|B), which is read as "the probability of A, given B."
The key phrase to look for in word problems is "given that," "if," or "of those." c) The Conditional Probability Formula The formal definition of conditional probability is: P(A|B) = P(A and B) / P(B)
Where: P(A|B) is the probability of event A happening, given that B has happened. P(A and B) (or P(A ∩ B)) is the probability of both A and B happening together. P(B) is the probability of event B happening.