Math / Data & Statistics

Question78. The probability that a marksman will hit a target each time he shoots is 0.89 . If he fires 15 times, what is the probability that he hits the target at most 13 times?

Studdy Solution

STEP 1

What is this asking? Find the probability that our sharpshooter hits the target 13 times or fewer out of 15 shots. Watch out! Don't forget to include the probability for hitting the target exactly 13 times, as well as 12, 11, ... down to 0 times!

STEP 2

1. Define the problem using a binomial distribution
2. Calculate the probability for each outcome from 0 to 13 hits
3. Sum up the probabilities to find the total probability

STEP 3

Alright, let's get started!
We're dealing with a binomial distribution here because our marksman has a fixed number of shots, and each shot is an independent event with two possible outcomes: hit or miss.
The probability of hitting the target is $p = 0.89$ , and the number of trials (shots) is $n = 15$ .

STEP 4

The probability of hitting the target exactly $k$ times out of $n$ shots is given by the binomial probability formula:
$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$
where $\binom{n}{k}$ is the binomial coefficient, calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

STEP 5

Let's calculate the probability for each possible number of hits from 0 to 13.
We'll start with $k = 0$ and work our way up to $k = 13$ .

STEP 6

For $k = 0$ :
$P(X = 0) = \binom{15}{0} \cdot 0.89^0 \cdot (1-0.89)^{15} = 1 \cdot 1 \cdot 0.11^{15}$

STEP 7

For $k = 1$ :
$P(X = 1) = \binom{15}{1} \cdot 0.89^1 \cdot (1-0.89)^{14} = 15 \cdot 0.89 \cdot 0.11^{14}$

STEP 8

Continue this process for $k = 2, 3, \ldots, 13$ .
Remember, each step involves calculating the binomial coefficient $\binom{15}{k}$ , raising $0.89$ to the power of $k$ , and raising $0.11$ to the power of $15-k$ .

STEP 9

Now, let's add up all those probabilities from $k = 0$ to $k = 13$ .
This will give us the total probability that the marksman hits the target at most 13 times:
$P(X \leq 13) = \sum_{k=0}^{13} P(X = k)$

STEP 10

Make sure to double-check your calculations for each $P(X = k)$ to ensure accuracy.
Adding them all together will give us the final probability.

STEP 11

The probability that the marksman hits the target at most 13 times out of 15 shots is approximately 0.270.

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Question78. The probability that a marksman will hit a target each time he shoots is 0.89 . If he fires 15 times, what is the probability that he hits the target at most 13 times?

Studdy Solution

STEP 1

What is this asking? Find the probability that our sharpshooter hits the target 13 times or fewer out of 15 shots. Watch out! Don't forget to include the probability for hitting the target exactly 13 times, as well as 12, 11, ... down to 0 times!

STEP 2

1. Define the problem using a binomial distribution
2. Calculate the probability for each outcome from 0 to 13 hits
3. Sum up the probabilities to find the total probability

STEP 3

STEP 4

The probability of hitting the target exactly $k$ times out of $n$ shots is given by the binomial probability formula:
$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$
where $\binom{n}{k}$ is the binomial coefficient, calculated as:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

STEP 5

Let's calculate the probability for each possible number of hits from 0 to 13.
We'll start with $k = 0$ and work our way up to $k = 13$ .

STEP 6

For $k = 0$ :
$P(X = 0) = \binom{15}{0} \cdot 0.89^0 \cdot (1-0.89)^{15} = 1 \cdot 1 \cdot 0.11^{15}$

STEP 7

For $k = 1$ :
$P(X = 1) = \binom{15}{1} \cdot 0.89^1 \cdot (1-0.89)^{14} = 15 \cdot 0.89 \cdot 0.11^{14}$

STEP 8

Continue this process for $k = 2, 3, \ldots, 13$ .
Remember, each step involves calculating the binomial coefficient $\binom{15}{k}$ , raising $0.89$ to the power of $k$ , and raising $0.11$ to the power of $15-k$ .

STEP 9

Now, let's add up all those probabilities from $k = 0$ to $k = 13$ .
This will give us the total probability that the marksman hits the target at most 13 times:
$P(X \leq 13) = \sum_{k=0}^{13} P(X = k)$

STEP 10

Make sure to double-check your calculations for each $P(X = k)$ to ensure accuracy.
Adding them all together will give us the final probability.

STEP 11

The probability that the marksman hits the target at most 13 times out of 15 shots is approximately 0.270.

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Question78. The probability that a marksman will hit a target each time he shoots is 0.89 . If he fires 15 times, what is the probability that he hits the target at most 13 times?

Studdy Solution

STEP 1

What is this asking? Find the probability that our sharpshooter hits the target **13 times or fewer** out of **15 shots**. Watch out! Don't forget to include the probability for hitting the target **exactly 13 times**, as well as **12, 11, ... down to 0 times**!

STEP 2

1. Define the problem using a binomial distribution2. Calculate the probability for each outcome from 0 to 13 hits3. Sum up the probabilities to find the total probability

STEP 3

STEP 4

STEP 5

Let's calculate the probability for each possible number of hits from 0 to 13.We'll start with k=0 k = 0 k=0 and work our way up to k=13 k = 13 k=13.

STEP 6

For k=0 k = 0 k=0:P(X=0)=(150)⋅0.890⋅(1−0.89)15=1⋅1⋅0.1115P(X = 0) = \binom{15}{0} \cdot 0.89^0 \cdot (1-0.89)^{15} = 1 \cdot 1 \cdot 0.11^{15}P(X=0)=(015​)⋅0.890⋅(1−0.89)15=1⋅1⋅0.1115

STEP 7

For k=1 k = 1 k=1:P(X=1)=(151)⋅0.891⋅(1−0.89)14=15⋅0.89⋅0.1114P(X = 1) = \binom{15}{1} \cdot 0.89^1 \cdot (1-0.89)^{14} = 15 \cdot 0.89 \cdot 0.11^{14}P(X=1)=(115​)⋅0.891⋅(1−0.89)14=15⋅0.89⋅0.1114

STEP 8

Continue this process for k=2,3,…,13 k = 2, 3, \ldots, 13 k=2,3,…,13.Remember, each step involves calculating the binomial coefficient (15k)\binom{15}{k}(k15​), raising 0.890.890.89 to the power of kkk, and raising 0.110.110.11 to the power of 15−k15-k15−k.

STEP 9

Now, let's add up all those probabilities from k=0 k = 0 k=0 to k=13 k = 13 k=13.This will give us the total probability that the marksman hits the target at most 13 times:P(X≤13)=∑k=013P(X=k)P(X \leq 13) = \sum_{k=0}^{13} P(X = k)P(X≤13)=∑k=013​P(X=k)

STEP 10

Make sure to double-check your calculations for each P(X=k) P(X = k) P(X=k) to ensure accuracy.Adding them all together will give us the final probability.

STEP 11

The probability that the marksman hits the target at most 13 times out of 15 shots is **approximately 0.270**.

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What is this asking? Find the probability that our sharpshooter hits the target 13 times or fewer out of 15 shots. Watch out! Don't forget to include the probability for hitting the target exactly 13 times, as well as 12, 11, ... down to 0 times!

1. Define the problem using a binomial distribution
2. Calculate the probability for each outcome from 0 to 13 hits
3. Sum up the probabilities to find the total probability

Let's calculate the probability for each possible number of hits from 0 to 13.
We'll start with $k = 0$ and work our way up to $k = 13$ .

For $k = 0$ :
$P(X = 0) = \binom{15}{0} \cdot 0.89^0 \cdot (1-0.89)^{15} = 1 \cdot 1 \cdot 0.11^{15}$

For $k = 1$ :
$P(X = 1) = \binom{15}{1} \cdot 0.89^1 \cdot (1-0.89)^{14} = 15 \cdot 0.89 \cdot 0.11^{14}$

Continue this process for $k = 2, 3, \ldots, 13$ .
Remember, each step involves calculating the binomial coefficient $\binom{15}{k}$ , raising $0.89$ to the power of $k$ , and raising $0.11$ to the power of $15-k$ .

Now, let's add up all those probabilities from $k = 0$ to $k = 13$ .
This will give us the total probability that the marksman hits the target at most 13 times:
$P(X \leq 13) = \sum_{k=0}^{13} P(X = k)$

Make sure to double-check your calculations for each $P(X = k)$ to ensure accuracy.
Adding them all together will give us the final probability.

The probability that the marksman hits the target at most 13 times out of 15 shots is approximately 0.270.