A) The student will compare and contrast the probability of independent and dependent events.
B) The student will determine probabilities for independent and dependent events.
Key Terms
Event: when something occurs.
Examples: flipping a coin, rolling a die, spinning a spinner, picking a marble out of a bag.
Simple Event: when only a single event happens.
Compound Event: when two or more events happen at the same time (or one right after the other).
Independent Event: when the events don't impact / affect each other.
Replacement: when something is chosen, put back, and has the chance to be picked again.
Dependent Event: when the events do impact / affect each other.
Without Replacement: when something is chosen, set aside, and does not have the chance to be picked again. This changes the number of choices available, which impacts the second event.
Outcome: the result of an event.
Desired Outcome: the result that you want to happen.
Sample Space: the set (group) of all possible outcomes / results (important for calculating probability).
Probability: how likely something is to occur.
Impossible: something has a 0% chance of happening / is guaranteed to not happen.
Not Likely: when something has less than a 50% chance of happening.
Equally Likely: when outcomes have an equal chance of happening (doesn't matter how many outcomes are possible).
Example:
Likely: when something has more than a 50% chance of happening.
Certain: something has a 100% chance of happening / is guaranteed to happen.
Important Info:
The probability of an event (outcome) occurring can be represented as a ratio of the desired outcome (what you want to happen) compared to the sample space (all possible outcomes). This ratio can be expressed as a fraction, decimal, or percent.
Example: you want to flip heads on a coin. The chances of this are 1 (heads) out of 2 (heads, tails). This can be written as 1/2, 0.5 (when you do 1 ÷ 2), or as 50% (convert the decimal to a percent).
The probability of an event occurring is a ratio between 0 and 1.
A probability of zero means the event will never occur. #impossible
A probability of one means the event will always occur. #certain
Think of it in terms of how much pizza you eat: if you ate 0% of the pizza, you didn't eat any of it. If you ate 100% of a pizza, you ate "1 whole" pizza (1 whole = 1).
There are simple events and compound events.
Simple events are when a single event occurs (keep it simple lol).
Compound events are when 2 or more events occur (e.g., rolling a number cube and flipping a coin, or pulling 2 marbles out of a bag). Compound events can be independent or dependent.
Independent events do not impact each other (flipping a coin, rolling a number cube, etc.). Independent events also occur when you choose an item and put it back before making a second choice. This is called (with) replacement.
Dependent events occur when the first event impacts the second event. This usually occurs when something is chosen, and then a second item is chosen without putting the first item back. This is called without replacement.
Although dependent events usually involve choosing something and not putting it back, that's not always the case (99% of the time it will be). An example of this not being the case is when the chance that school is closed is directly impacted by the chance of snow for that day.
The probability of an event happening is written as P(A) where "A" is the event. Some examples:
When flipping a coin, P(heads) = 1/2
When rolling a number cube, P(3) is 1/6
When choosing a letter, P(vowel) = 5/26
Sometimes we are happy with more than one outcome. An example would be picking candy out of a bucket. I like Snickers and Nestle Crunch but not Mounds.
If there are 5 Snickers, 3 Nestle Crunch, and 2 Mounds, I want the probability of choosing a Snickers or Nestle Crunch. I don't care which one, as long as I don't choose Mounds. Here's how to calculate that:
P(Snickers or Nestle Crunch): (5+3)/10 = 8/10 = 4/5
When finding the probability of two events, it's a little different.
For independent events, the probability is written as P(A and B) where "A" and "B" are the desirable outcomes. To find the probability of both outcomes occurring, we multiply them: P(A)∙P(B). See this linked pic for an understanding why.
For dependent events, the probability is written as P(A then B) where "A" and "B" are the desirable outcomes. This is different from independent events because we've chosen an item, and then chosen a second item without putting the first one back:
Using my Snickers / Nestle Crunch / Mounds example from above, let's find the chance I'd pick both Mounds.
P(Mounds then Mounds) = 2/10 for the first one and then 1/9 for the second one (we're pretending that I picked the first Mounds bar.
P(3 and heads) = 16∙12=112
Fundamental counting principle
If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. For example, if you choose a blue card from a set of nine different colored cards that has a total of four blue cards and you do not place that blue card back in the set before selecting a second card, the chance of selecting a blue card the second time is diminished because there are now only three blue cards remaining in the set. Other examples of dependent events include, but are not limited to: choosing two marbles from a bag but not replacing the first after selecting it; determining the probability that it will snow and that school will be cancelled.
The probability of two dependent events is found by using the following formula: P(A and B) = P(A)∙P(B after A)
- Example: You have a bag holding a blue ball, a red ball, and a yellow ball. What is the probability of picking a blue ball out of the bag on the first pick then without replacing the blue ball in the bag, picking a red ball on the second pick?
Determine whether two events are independent or dependent. (a)
Compare and contrast the probability of independent and dependent events. (a)
Determine the probability of two independent events. (b)
Determine the probability of two dependent events. (b)