Question 1171394: 27 % adults favor the use of unmanned drones by police agencies. Twelve U.S. adults are randomly selected. Find the probability that the number of U.S. adults who favor the use of unmanned drones by police agencies is (a) exactly three, (b) at least four, (c) less than eight.
(a) P(3)=
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let $X$ be the number of U.S. adults who favor the use of unmanned drones by police agencies.
We are given that the probability of an adult favoring the use of drones is $p = 0.27$.
We have a sample of $n = 12$ adults.
This is a binomial distribution, so $X \sim B(12, 0.27)$.
The probability mass function is given by:
$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$
(a) We want to find $P(X = 3)$, so $k = 3$:
$$ P(X = 3) = \binom{12}{3} (0.27)^3 (1-0.27)^{12-3} $$
$$ P(X = 3) = \binom{12}{3} (0.27)^3 (0.73)^9 $$
$$ \binom{12}{3} = \frac{12!}{3!9!} = \frac{12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1} = 2 \cdot 11 \cdot 10 = 220 $$
$$ P(X = 3) = 220 (0.27)^3 (0.73)^9 $$
$$ P(X = 3) \approx 220 (0.019683) (0.066373) $$
$$ P(X = 3) \approx 220 (0.0013063) \approx 0.287386 $$
$$ P(X = 3) \approx 0.2874 $$
(b) We want to find $P(X \ge 4)$, which is $1 - P(X < 4) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]$.
$$ P(X = 0) = \binom{12}{0} (0.27)^0 (0.73)^{12} \approx 1(1)(0.033626) \approx 0.0336 $$
$$ P(X = 1) = \binom{12}{1} (0.27)^1 (0.73)^{11} \approx 12(0.27)(0.046063) \approx 0.1492 $$
$$ P(X = 2) = \binom{12}{2} (0.27)^2 (0.73)^{10} \approx 66(0.0729)(0.0631) \approx 0.3039 $$
$$ P(X < 4) = 0.0336 + 0.1492 + 0.3039 + 0.2874 = 0.7741 $$
$$ P(X \ge 4) = 1 - 0.7741 = 0.2259 $$
(c) We want to find $P(X < 8) = 1 - P(X \ge 8) = 1 - [P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)]$.
Alternatively, we can compute $P(X < 8) = \sum_{k=0}^{7} P(X=k)$.
Using a calculator or software:
$$ P(X < 8) \approx 0.9997 $$
Using the complement, $P(X \ge 8)$:
$$ P(X=8) \approx 0.00021 $$
$$ P(X=9) \approx 0.00002 $$
$$ P(X=10) \approx 0.000001 $$
$$ P(X=11) \approx 0.00000003 $$
$$ P(X=12) \approx 0.0000000002 $$
Summing those gives $P(X\ge 8) \approx 0.00023$.
$P(X<8) = 1 - 0.00023 = 0.99977 \approx 0.9998$.
Final Answer: The final answer is $\boxed{0.2874, 0.2259, 0.9998}$
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
27 % adults favor the use of unmanned drones by police agencies.
Twelve U.S. adults are randomly selected. Find the probability
that the number of U.S. adults who favor the use of unmanned drones by police agencies is
(a) exactly three,
(b) at least four,
(c) less than eight.
(a) P(3)=
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Be aware !
In the post by @CPhill, all calculations are INCORRECT.
The correct answers are
(a) 0.2549
(b) 0.4138
(c) 0.9953
Use online free of charge calculator
https://stattrek.com/online-calculator/binomial.aspx
to check your calculations.
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Regarding the post by @CPhill . . .
Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.
The artificial intelligence is like a baby now. It is in the experimental stage
of development and can make mistakes and produce nonsense without any embarrassment.
It has no feeling of shame - it is shameless.
This time, again, it made an error.
Although the @CPhill' solutions are copy-paste Google AI solutions, there is one essential difference.
Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.
All @CPhill' solutions are copy-paste of Google AI solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.
Every time, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.
And the last my comment.
When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.
Without it, their reliability is ZERO and their creadability is ZERO, too.
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