Outcome and Events

Probability of Independent Events

By DANIEL SOLOMON

Math for fun

Emma and her mom are playing a number cube game and it is Emma’s turn to roll the number cube. If she can roll a 5, she will win the game. Emma believes that the chances of rolling a 5 are pretty small—She thinks it's only 1 in 6.​

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1- Explain, using words and drawings whether Emma is correct.

2- Write P(5) in as many ways as you can​.

Draw

1- Explain, using words and diagrams whether Emma is correct.

2- Write P(5) in as many ways as you can​.

​Refresher:​

Probability is the chance of something specific happening from a list of possible events that could happen.

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The desired event is called the favorable outcome.

All the possible events are called the possible outcomes.

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Probability can be expressed with a fraction where the number on top (numerator) is the number of favorable outcomes and the number on the bottom (denominator) is the number of all possible outcomes.

P = number of ways favorable outcomes could happen

number of all possible outcomes

Probability of Independent events

​Events are called independent events if they do not affect each other.​

Which means one does not need the other to happen.​

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​Probability of independent events when they happen at the same time. Example:​

John rolls a six-sided number cube at the same time that Ahmed flips a coin.

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Determine all possible outcomes if they complete the task at the exact same time.

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Organize the possible outcomes anyway you can.​

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Rolling the number cube and flipping the coin are two independent events so they don't affect each other, and when they happen at the same time​, all the possible outcomes can be represented with a tree diagram.

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Using a Tree diagram

Or using a table

Using a table

Open Ended

What do you notice about the total number of outcomes when two independent events happen at the same time ?

Remember, John has 6 possible outcomes and Ahmed has 2 possible outcomes.

Poll

So far, I've understood that

• $$P=\frac{\left(number\ of\ favorable\ outomes\right)}{total\ number\ of\ outcomes}$$  

• Independent events are events that do not affect each other

• When independent events happen at the same time, the total number of outcomes equals the product of outcomes of both events

• I can represent all the outcomes using a tree diagram or a table

• None of the above

Given the previous example, determine

P( H, 3)

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Probability of independent events when they happen at the same time

Given the previous example, determine

​​P( H, even number)

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Probability of independent events when they happen at the same time

Given the previous example, determine

​​P(H, 1 Or 2)

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Probability of independent events when they happen at the same time

​P = number of ways favorable outcomes could happen

number of all possible outcomes

​Recap:

Independent events are events that don't affect each other.

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if A and B are independent events:

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​Probability of A and B happening at the same time =​ P(A) * P(B)

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Jackson spins a three-sectioned spinner with the letters R, U, and N, and Roberta flips a nickel.

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Determine all possible outcomes if Jackson and Roberta complete the task at the same time.

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Practice:

​Determine

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P(R, Heads)

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P(consonant, T). ​

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Practice:

​You flip a coin and spin a five-sectioned spinner that has the colours red, yellow, blue, green, and orange on it.

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Determine P(heads and red).

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Verify your answer with a tree diagram or a table.​

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Practice:

What is the probability of landing on tails for a toss coin and rolling a 2 on a six-sided number cube at the same time?

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Verify your answer with a tree diagram or a table.​

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Practice:

You roll a regular number cube and spin a four-sectioned spinner with the letters T, R, E, and E on it.

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Determine P(4 and E).

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Verify your answer with a tree diagram or a table.​

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Practice: