Solved on Feb 08, 2024

Find $P(X=19)$ for the given sampling distribution: $X = \{-16, -12, -7, 10, 19\}$ and $P(X)= \{1/100, 1/50, 9/100, 3/50, \text{?}\}$ .

STEP 1

Assumptions
1. The values of $X$ represent the possible outcomes of a random variable.
2. $P(X)$ represents the probability of each outcome.
3. The probabilities of all possible outcomes must sum up to 1 because the total probability of all outcomes in a probability distribution is always 1.

STEP 2

Calculate the sum of the given probabilities.
$\sum P(X) = \frac{1}{100} + \frac{1}{50} + \frac{9}{100} + \frac{3}{50}$

STEP 3

Find a common denominator to add the fractions. The common denominator for 100 and 50 is 100.
$\frac{1}{100} = \frac{1}{100}$ $\frac{1}{50} = \frac{2}{100}$ $\frac{9}{100} = \frac{9}{100}$ $\frac{3}{50} = \frac{6}{100}$

STEP 4

Add the fractions with the common denominator.
$\sum P(X) = \frac{1}{100} + \frac{2}{100} + \frac{9}{100} + \frac{6}{100}$

STEP 5

Perform the addition.
$\sum P(X) = \frac{1 + 2 + 9 + 6}{100}$ $\sum P(X) = \frac{18}{100}$

STEP 6

Simplify the fraction.
$\sum P(X) = \frac{18}{100} = \frac{9}{50}$

STEP 7

Subtract the sum of the given probabilities from 1 to find the missing probability $P(X=19)$ .
$P(X=19) = 1 - \sum P(X)$

STEP 8

Plug in the calculated sum of probabilities into the equation.
$P(X=19) = 1 - \frac{9}{50}$

STEP 9

Find a common denominator to subtract the fractions. The common denominator for 1 and 50 is 50.
$1 = \frac{50}{50}$

STEP 10

Perform the subtraction.
$P(X=19) = \frac{50}{50} - \frac{9}{50}$

STEP 11

Calculate the result.
$P(X=19) = \frac{50 - 9}{50}$ $P(X=19) = \frac{41}{50}$

STEP 12

Convert the fraction to a decimal to match the answer choices.
$P(X=19) = \frac{41}{50} = 0.82$
The probability $P(X=19)$ is 0.82.

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Find $P(X=19)$ for the given sampling distribution: $X = \{-16, -12, -7, 10, 19\}$ and $P(X)= \{1/100, 1/50, 9/100, 3/50, \text{?}\}$ .

STEP 1

Assumptions
1. The values of $X$ represent the possible outcomes of a random variable.
2. $P(X)$ represents the probability of each outcome.
3. The probabilities of all possible outcomes must sum up to 1 because the total probability of all outcomes in a probability distribution is always 1.

STEP 2

Calculate the sum of the given probabilities.
$\sum P(X) = \frac{1}{100} + \frac{1}{50} + \frac{9}{100} + \frac{3}{50}$

STEP 3

Find a common denominator to add the fractions. The common denominator for 100 and 50 is 100.
$\frac{1}{100} = \frac{1}{100}$ $\frac{1}{50} = \frac{2}{100}$ $\frac{9}{100} = \frac{9}{100}$ $\frac{3}{50} = \frac{6}{100}$

STEP 4

Add the fractions with the common denominator.
$\sum P(X) = \frac{1}{100} + \frac{2}{100} + \frac{9}{100} + \frac{6}{100}$

STEP 5

Perform the addition.
$\sum P(X) = \frac{1 + 2 + 9 + 6}{100}$ $\sum P(X) = \frac{18}{100}$

STEP 6

Simplify the fraction.
$\sum P(X) = \frac{18}{100} = \frac{9}{50}$

STEP 7

Subtract the sum of the given probabilities from 1 to find the missing probability $P(X=19)$ .
$P(X=19) = 1 - \sum P(X)$

STEP 8

Plug in the calculated sum of probabilities into the equation.
$P(X=19) = 1 - \frac{9}{50}$

STEP 9

Find a common denominator to subtract the fractions. The common denominator for 1 and 50 is 50.
$1 = \frac{50}{50}$

STEP 10

Perform the subtraction.
$P(X=19) = \frac{50}{50} - \frac{9}{50}$

STEP 11

Calculate the result.
$P(X=19) = \frac{50 - 9}{50}$ $P(X=19) = \frac{41}{50}$

STEP 12

Convert the fraction to a decimal to match the answer choices.
$P(X=19) = \frac{41}{50} = 0.82$
The probability $P(X=19)$ is 0.82.

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Find P(X=19)P(X=19)P(X=19) for the given sampling distribution: X={−16,−12,−7,10,19}X = \{-16, -12, -7, 10, 19\}X={−16,−12,−7,10,19} and P(X)={1/100,1/50,9/100,3/50,?}P(X)= \{1/100, 1/50, 9/100, 3/50, \text{?}\}P(X)={1/100,1/50,9/100,3/50,?}.

STEP 1

Assumptions1. The values of XXX represent the possible outcomes of a random variable.2. P(X)P(X)P(X) represents the probability of each outcome.3. The probabilities of all possible outcomes must sum up to 1 because the total probability of all outcomes in a probability distribution is always 1.

STEP 2

Calculate the sum of the given probabilities.∑P(X)=1100+150+9100+350\sum P(X) = \frac{1}{100} + \frac{1}{50} + \frac{9}{100} + \frac{3}{50}∑P(X)=1001​+501​+1009​+503​

STEP 3

STEP 4

Add the fractions with the common denominator.∑P(X)=1100+2100+9100+6100\sum P(X) = \frac{1}{100} + \frac{2}{100} + \frac{9}{100} + \frac{6}{100}∑P(X)=1001​+1002​+1009​+1006​

STEP 5

Perform the addition.∑P(X)=1+2+9+6100\sum P(X) = \frac{1 + 2 + 9 + 6}{100}∑P(X)=1001+2+9+6​ ∑P(X)=18100\sum P(X) = \frac{18}{100}∑P(X)=10018​

STEP 6

Simplify the fraction.∑P(X)=18100=950\sum P(X) = \frac{18}{100} = \frac{9}{50}∑P(X)=10018​=509​

STEP 7

Subtract the sum of the given probabilities from 1 to find the missing probability P(X=19)P(X=19)P(X=19).P(X=19)=1−∑P(X)P(X=19) = 1 - \sum P(X)P(X=19)=1−∑P(X)

STEP 8

Plug in the calculated sum of probabilities into the equation.P(X=19)=1−950P(X=19) = 1 - \frac{9}{50}P(X=19)=1−509​

STEP 9

Find a common denominator to subtract the fractions. The common denominator for 1 and 50 is 50.1=50501 = \frac{50}{50}1=5050​

STEP 10

Perform the subtraction.P(X=19)=5050−950P(X=19) = \frac{50}{50} - \frac{9}{50}P(X=19)=5050​−509​

STEP 11

Calculate the result.P(X=19)=50−950P(X=19) = \frac{50 - 9}{50}P(X=19)=5050−9​ P(X=19)=4150P(X=19) = \frac{41}{50}P(X=19)=5041​

STEP 12

Convert the fraction to a decimal to match the answer choices.P(X=19)=4150=0.82P(X=19) = \frac{41}{50} = 0.82P(X=19)=5041​=0.82The probability P(X=19)P(X=19)P(X=19) is 0.82.

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Find $P(X=19)$ for the given sampling distribution: $X = \{-16, -12, -7, 10, 19\}$ and $P(X)= \{1/100, 1/50, 9/100, 3/50, \text{?}\}$ .

Assumptions
1. The values of $X$ represent the possible outcomes of a random variable.
2. $P(X)$ represents the probability of each outcome.
3. The probabilities of all possible outcomes must sum up to 1 because the total probability of all outcomes in a probability distribution is always 1.

Calculate the sum of the given probabilities.
$\sum P(X) = \frac{1}{100} + \frac{1}{50} + \frac{9}{100} + \frac{3}{50}$

Add the fractions with the common denominator.
$\sum P(X) = \frac{1}{100} + \frac{2}{100} + \frac{9}{100} + \frac{6}{100}$

Perform the addition.
$\sum P(X) = \frac{1 + 2 + 9 + 6}{100}$ $\sum P(X) = \frac{18}{100}$

Simplify the fraction.
$\sum P(X) = \frac{18}{100} = \frac{9}{50}$

Subtract the sum of the given probabilities from 1 to find the missing probability $P(X=19)$ .
$P(X=19) = 1 - \sum P(X)$

Plug in the calculated sum of probabilities into the equation.
$P(X=19) = 1 - \frac{9}{50}$

Find a common denominator to subtract the fractions. The common denominator for 1 and 50 is 50.
$1 = \frac{50}{50}$

Perform the subtraction.
$P(X=19) = \frac{50}{50} - \frac{9}{50}$

Calculate the result.
$P(X=19) = \frac{50 - 9}{50}$ $P(X=19) = \frac{41}{50}$

Convert the fraction to a decimal to match the answer choices.
$P(X=19) = \frac{41}{50} = 0.82$
The probability $P(X=19)$ is 0.82.