Question

checking for independence
you know that if two events, a and b, are independent events, then
p(a and b)=p(a)×p(b). the converse is also true. if
p(c and d)=p(c)×p(d), then events c and d are independent events.
it is sometimes difficult to know whether two events are independent just by
considering the situation. so, you can check to see if two events are
independent by multiplying their probabilities. if the product is equal to the
probability of the two events happening together, then the events are
independent.
consider the cards shown. are the events \choosing a number card\ and
\choosing a circle card\ independent events?
study the worked example and then answer each question.
consider the cards in the example. are the events \choosing a letter card\ and
\choosing a square card\ independent events?
p(letter card)=
p(square card)=
p(letter and square)=
the two events, \choosing a letter card\ and \choosing a square card\ are
independent events.
a meteorologist says that there is an 80% chance of rain and a 50% chance of

Explanation:

Step1: Calculate \(P(\text{letter card})\)

There are 4 cards in total. The number of letter cards (\(A\) and \(B\)) is 2. So \(P(\text{letter card})=\frac{2}{4}=\frac{1}{2}\)

Step2: Calculate \(P(\text{square card})\)

The number of square - shaped cards (\(4\) and \(A\)) is 2. So \(P(\text{square card})=\frac{2}{4}=\frac{1}{2}\)

Step3: Calculate \(P(\text{letter and square})\)

The only card that is a letter and square - shaped is \(A\). So \(P(\text{letter and square})=\frac{1}{4}\)

Step4: Check independence

Since \(P(\text{letter card})\times P(\text{square card})=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}\) and \(P(\text{letter and square})=\frac{1}{4}\)

Answer:

\(P(\text{letter card})=\frac{1}{2}\)
\(P(\text{square card})=\frac{1}{2}\)
\(P(\text{letter and square})=\frac{1}{4}\)