Math / Data & Statistics

QuestionQuestion 6
In a large population, $55 \%$ of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?
Give your answer as a decimal to 4 places.

Studdy Solution

STEP 1

What is this asking? If we grab 5 random people, what are the chances that at least one of them got the vaccine? Watch out! "At least one" is not the same as "exactly one"!
It means one or more.

STEP 2

1. Find the probability that none are vaccinated.
2. Find the probability that at least one is vaccinated.

STEP 3

Since $55\%$ of people are vaccinated, that means $100\% - 55\% = 45\%$ are not vaccinated.
So the probability of one random person not being vaccinated is $45\%$ , or $0.45$ .

STEP 4

We're picking 5 people.
We want the probability that none of them are vaccinated.
Since each person is chosen randomly, the probability of each person not being vaccinated is independent of the others.
So, we can just multiply the individual probabilities together!

STEP 5

The probability that none of the 5 people are vaccinated is $0.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 = (0.45)^5$ .
Let's calculate that: $(0.45)^5 \approx 0.01845$ .

STEP 6

Remember, there are only two possibilities: either none of the people are vaccinated, or at least one is.
These two probabilities must add up to $1$ (or $100\%$ ).

STEP 7

We know the probability that none are vaccinated is approximately $0.01845$ .
So, the probability that at least one is vaccinated is $1 - 0.01845 = 0.98155$ .

STEP 8

The problem asks for 4 decimal places.
Rounding $0.98155$ to 4 decimal places gives us $0.9816$ .

STEP 9

The probability that at least one person is vaccinated is approximately $0.9816$ .

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QuestionQuestion 6
In a large population, $55 \%$ of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?
Give your answer as a decimal to 4 places.

Studdy Solution

STEP 1

What is this asking? If we grab 5 random people, what are the chances that at least one of them got the vaccine? Watch out! "At least one" is not the same as "exactly one"!
It means one or more.

STEP 2

1. Find the probability that none are vaccinated.
2. Find the probability that at least one is vaccinated.

STEP 3

Since $55\%$ of people are vaccinated, that means $100\% - 55\% = 45\%$ are not vaccinated.
So the probability of one random person not being vaccinated is $45\%$ , or $0.45$ .

STEP 4

We're picking 5 people.
We want the probability that none of them are vaccinated.
Since each person is chosen randomly, the probability of each person not being vaccinated is independent of the others.
So, we can just multiply the individual probabilities together!

STEP 5

The probability that none of the 5 people are vaccinated is $0.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 = (0.45)^5$ .
Let's calculate that: $(0.45)^5 \approx 0.01845$ .

STEP 6

Remember, there are only two possibilities: either none of the people are vaccinated, or at least one is.
These two probabilities must add up to $1$ (or $100\%$ ).

STEP 7

We know the probability that none are vaccinated is approximately $0.01845$ .
So, the probability that at least one is vaccinated is $1 - 0.01845 = 0.98155$ .

STEP 8

The problem asks for 4 decimal places.
Rounding $0.98155$ to 4 decimal places gives us $0.9816$ .

STEP 9

The probability that at least one person is vaccinated is approximately $0.9816$ .

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QuestionQuestion 6In a large population, 55%55 \%55% of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?Give your answer as a decimal to 4 places.

Studdy Solution

STEP 1

What is this asking? If we grab 5 random people, what are the chances that *at least one* of them got the vaccine? Watch out! "At least one" is not the same as "exactly one"!It means one *or more*.

STEP 2

1. Find the probability that *none* are vaccinated.2. Find the probability that *at least one* is vaccinated.

STEP 3

Since 55%55\%55% of people *are* vaccinated, that means 100%−55%=45%100\% - 55\% = 45\%100%−55%=45% are *not* vaccinated.So the probability of one random person *not* being vaccinated is 45%45\%45%, or 0.450.450.45.

STEP 4

We're picking 5 people.We want the probability that *none* of them are vaccinated.Since each person is chosen randomly, the probability of each person not being vaccinated is independent of the others.So, we can just multiply the individual probabilities together!

STEP 5

The probability that *none* of the 5 people are vaccinated is 0.45⋅0.45⋅0.45⋅0.45⋅0.45=(0.45)50.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 = (0.45)^50.45⋅0.45⋅0.45⋅0.45⋅0.45=(0.45)5.Let's calculate that: (0.45)5≈0.01845(0.45)^5 \approx 0.01845(0.45)5≈0.01845.

STEP 6

Remember, there are only two possibilities: either *none* of the people are vaccinated, or *at least one* is.These two probabilities *must* add up to 111 (or 100%100\%100%).

STEP 7

We know the probability that *none* are vaccinated is approximately 0.018450.018450.01845.So, the probability that *at least one* is vaccinated is 1−0.01845=0.981551 - 0.01845 = 0.981551−0.01845=0.98155.

STEP 8

The problem asks for 4 decimal places.Rounding 0.981550.981550.98155 to 4 decimal places gives us 0.98160.98160.9816.

STEP 9

The probability that at least one person is vaccinated is approximately 0.98160.98160.9816.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionQuestion 6
In a large population, $55 \%$ of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?
Give your answer as a decimal to 4 places.

What is this asking? If we grab 5 random people, what are the chances that at least one of them got the vaccine? Watch out! "At least one" is not the same as "exactly one"!
It means one or more.

1. Find the probability that none are vaccinated.
2. Find the probability that at least one is vaccinated.

Since $55\%$ of people are vaccinated, that means $100\% - 55\% = 45\%$ are not vaccinated.
So the probability of one random person not being vaccinated is $45\%$ , or $0.45$ .

We're picking 5 people.
We want the probability that none of them are vaccinated.
Since each person is chosen randomly, the probability of each person not being vaccinated is independent of the others.
So, we can just multiply the individual probabilities together!

The probability that none of the 5 people are vaccinated is $0.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 \cdot 0.45 = (0.45)^5$ .
Let's calculate that: $(0.45)^5 \approx 0.01845$ .

Remember, there are only two possibilities: either none of the people are vaccinated, or at least one is.
These two probabilities must add up to $1$ (or $100\%$ ).

We know the probability that none are vaccinated is approximately $0.01845$ .
So, the probability that at least one is vaccinated is $1 - 0.01845 = 0.98155$ .

The problem asks for 4 decimal places.
Rounding $0.98155$ to 4 decimal places gives us $0.9816$ .

The probability that at least one person is vaccinated is approximately $0.9816$ .