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A die is to be rolled and we are to observe the number that falls face up. a.) which pairs of the...

Question:

A die is to be rolled and we are to observe the number that falls face up.

a.) which pairs of the event (A&B, B&C, and A&C) are independent?

b.) Which pairs of events (A&B, B&C, and A&C) are mutually exclusive? How do I test these mathematically?

Event A: Observe a six - 1/6

Event B: Observe an odd number: 1/2

Event C: Observe a number greater than 3: 1/2

Mutually Exclusive Events:

Two events A and B that are mutually exclusive, do not share any element in common. That is:

{eq}P(A \cap B) = 0 {/eq}

And two X and Y events are independent when they satisfy the following property:

{eq}P(X \cap Y) = P(X) \times P(Y) {/eq}

Most importantly, being mutually exclusive and being independent are two different things.

Answer and Explanation:


Given Information:

A die is rolled. Consider the following events:

  • Event A: Observe a six - P(A) = 1/6
  • Event B: Observe an odd number - P(B) = 1/2
  • Event C: Observe a number greater than 3 - P(C) = 1/2


a.)

The probability for the event (A&B), that is

{eq}P(A \cap B) = 0 {/eq}

Since a six is not an odd number.


The probability for the event (B&C) is:

{eq}P(B \cap C) = \dfrac{1}{6} {/eq}

Sincethe only number that is greater than 3 and is odd is 5.


The probability for the event (A&C) is:

{eq}P(A \cap C) = \dfrac{1}{6} {/eq}

Since the only common element of A and C is 6.


Two events X and Y, that are independent satisfy the following equation:

{eq}P(X \cap Y) = P(X) \times P(Y) {/eq}

Since, none of the events satisfy the given equation, thus, none of the events are independent.


b.)

The events A and B are mutually exclusive since they do not have any element in common.

To check if two events X and Y are mutually exclusive, they must satisfy the following equations:

{eq}\begin{align*} P(X \cap Y) &= 0\\ P(X \cup Y) &= P(X) + P(Y) \end{align*} {/eq}


Learn more about this topic:

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Mutually Exclusive in Statistics: Definition, Formula & Examples

from High School Algebra II: Help and Review

Chapter 25 / Lesson 8
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