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You roll a $6$ -sided die two times. $\newline$ What is the probability of rolling a number less than $5$ and then rolling a number less than $2$ ? $\newline$ Simplify your answer and write it as a fraction or whole number. $\newline$ _____

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Probability of independent and dependent events

Full solution

Q. You roll a

6

-sided die two times.

\newline

What is the probability of rolling a number less than

5

and then rolling a number less than

2

?

\newline

Simplify your answer and write it as a fraction or whole number.

\newline

_____

Determine Probability of Rolling: First, we need to determine the probability of rolling a number less than $5$ on a $6$ -sided die. Numbers less than $5$ are $1$ , $2$ , $3$ , and $4$ . So there are $4$ favorable outcomes out of $6$ possible outcomes. $\newline$ Calculation: Probability of rolling a number less than $5 = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{6} = \frac{2}{3}.$
Find Probability of Second Roll: Next, we need to determine the probability of rolling a number less than $2$ on the second roll. The only number less than $2$ is $1$ , so there is $1$ favorable outcome out of $6$ possible outcomes. $\newline$ Calculation: Probability of rolling a number less than $2 = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{6}.$
Calculate Combined Probability: Now, we need to find the combined probability of both events happening in sequence. Since the rolls are independent events, we multiply the probabilities of each event. $\newline$ Calculation: Combined probability = Probability of first event $\times$ Probability of second event = $(\frac{2}{3}) \times (\frac{1}{6})$ .
Perform Multiplication: Perform the multiplication to find the combined probability. $\newline$ Calculation: Combined probability = $(\frac{2}{3}) \times (\frac{1}{6}) = \frac{2}{18} = \frac{1}{9}$ .

More problems from Probability of independent and dependent events

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college students running for student government class president. The candidates include

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education majors.

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of the candidates are randomly chosen to give the first

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speeches, what is the probability that all of them are education majors?

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Write your answer as a decimal rounded to four decimal places.

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Question

In an experiment, the probability that event

A

\frac{2}{7}

, the probability that event

B

\frac{7}{9}

, and the probability that events

A

B

\frac{1}{9}

\newline

A

B

independent events?

\newline

\newline

\text{yes}

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\text{no}

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Question

At a science museum, visitors can compete to see who has a faster reaction time. Competitors watch a red screen, and the moment they see it turn from red to green, they push a button. The machine records their reaction times and also asks competitors to report their gender.

\newline

The probability that a competitor reacted in over

0.7

0.6

, the probability that a competitor was female is

0.4

, and the probability that a competitor reacted in over

0.7

seconds and was female is

0.3

\newline

What is the probability that a randomly chosen competitor reacted in over

0.7

seconds or was female?

\newline

Write your answer as a whole number, decimal, or simplified fraction.

\newline

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Question

A restaurant server claims that \(28\)% of the time he leaves candy with the bill, the customer tips well.

\newline

If the server's claim is true, and one day he leaves candy with \(4\) customers' bills, what is the probability that exactly \(3\) of those customers will tip well?

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Write your answer as a decimal rounded to the nearest thousandth.

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Question

There is a spinner with `50` equally likely sections, numbered from `1` to `50`. You have the opportunity to spin it. If the number is odd, you win $`11`. If the number is even, you win nothing. If you play the game, what is the expected payoff?\(\$\)_____

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Question

There are two different raffles you can enter.

\newline

In raffle A, one ticket will win a `$720` prize, and the other tickets will win nothing. There are

1,000

in the raffle, each costing `$11`.

\newline

Raffle B is for a `$370` prize. Out of

125

tickets, each costing `$5`, one ticket will win the prize, and the other tickets will win nothing.

\newline

Which raffle is a better deal?

\newline

\newline

\text{[A]Raffle A}

\newline

\text{[B]Raffle B}

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Question

X

is a normally distributed random variable with mean

49

and standard deviation

25

\newline

What is the probability that

X

24

74

?

\newline

0.68-0.95-0.997

rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

\newline

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Question

X

is a normally distributed random variable with mean

65

and standard deviation

8

\newline

What is the probability that

X

62

68

?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

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Question

Complete the statement. Round your answer to the nearest thousandth.

\newline

In a population that is normally distributed with mean

43

and standard deviation

3

30\%

of the values are those less than ______.

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