# Binomial distribution.

#### Apprentice123

In a study of recognition of a mark, 95% of consumers recognize the brand. But beyond 15 randomly selected consumers, only 10 recognize the brand.

What is the probability of obtaining a maximum of 10 consumers recognize the brand among the 15 selected?

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#### ANDS!

What does "beyond 15 randomly selected. . ." mean? Does that mean someone asked 15 people and only 10 people recognized the brand? Not familiar with the wording.

If you assume that the population proportion is .95 recognize and .05 do not recognize, then this is a simple binomial distribution where you are calculating the probability of getting exactly 10 successes out of 15 tries. If you need help further than that, be sure to note where you are getting confused.

#### Apprentice123

What does "beyond 15 randomly selected. . ." mean? Does that mean someone asked 15 people and only 10 people recognized the brand? Not familiar with the wording.

If you assume that the population proportion is .95 recognize and .05 do not recognize, then this is a simple binomial distribution where you are calculating the probability of getting exactly 10 successes out of 15 tries. If you need help further than that, be sure to note where you are getting confused.
In a brand awareness 95% of consumers recognize the brand name "ABC".
However, among 15 randomly selected consumers only 10 recognize the brand name "ABC".
What is the probability of obtaining 10 consumers recognize the brand name "ABC" of the 15 selected?

#### undefined

MHF Hall of Honor
In a brand awareness 95% of consumers recognize the brand name "ABC".
However, among 15 randomly selected consumers only 10 recognize the brand name "ABC".
What is the probability of obtaining 10 consumers recognize the brand name "ABC" of the 15 selected?
From this wording it seems we want the probability that exactly 10 consumers recognize the brand out of 15 randomly selected, rather than at most 10.

If we assume an infinitely large population, then when picking out random consumers without replacement, the event that one consumer recognizes "ABC" is independent from the event that another consumer recognizes "ABC," always with probability 0.95.

So the probability we're after is

$$\displaystyle P(X = 10) = \binom{15}{10}(1-0.95)^{10}(0.95)^5$$.

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#### Apprentice123

From this wording it seems we want the probability that exactly 10 consumers recognize the brand out of 15 randomly selected, rather that at most 10.

If we assume an infinitely large population, then when picking out random consumers without replacement, the event that one consumer recognizes "ABC" is independent from the event that another consumer recognizes "ABC," always with probability 0.95.

So the probability we're after is

$$\displaystyle P(X = 10) = \binom{15}{10}(1-0.95)^{10}(0.95)^5$$.

But the answer is $$\displaystyle 0,0006146$$ I not find

#### undefined

MHF Hall of Honor
But the answer is $$\displaystyle 0,0006146$$ I not find
I don't get that answer either. I believe my method is correct. My guess (call it arrogance if you like) is that either the book is wrong, or not properly worded. I'm assuming you typed it out exactly as written. Anyone else care weigh in on this?

#### Apprentice123

I don't get that answer either. I believe my method is correct. My guess (call it arrogance if you like) is that either the book is wrong, or not properly worded. I'm assuming you typed it out exactly as written. Anyone else care weigh in on this?
My interpretation of the problem:
- 95% of consumers know "ABC"
- We selected 15 consumers
- Of these 15, only 10 known "ABC"

The question is:
- Of these 15, what is the probability of 10 consumers know "ABC" ?

#### undefined

MHF Hall of Honor
My interpretation of the problem:
- 95% of consumers know "ABC"
- We selected 15 consumers
- Of these 15, only 10 known "ABC"

The question is:
- Of these 15, what is the probability of 10 consumers know "ABC" ?
Not sure I follow you on that. If it is known that 10 of the 15 consumers know "ABC," then the probability that 10 of the 15 consumers know "ABC" is 1, or 100%.

#### Apprentice123

Not sure I follow you on that. If it is known that 10 of the 15 consumers know "ABC," then the probability that 10 of the 15 consumers know "ABC" is 1, or 100%.
Sorry. The question is:

Of the 15 selected, what is the probability of have at MAXIMUM 10 consumers know "ABC" ?

#### undefined

MHF Hall of Honor
Sorry. The question is:

Of the 15 selected, what is the probability of have at MAXIMUM 10 consumers know "ABC" ?
No problem! I get the book answer now.

Let X be the random variable: how many people of the 15 know "ABC."

We have $$\displaystyle P(X\leq 10) = 1 - P(X \geq 11) = 1 - P(X = 11) - \cdots - P(X = 15)$$

$$\displaystyle = 1 - \binom{15}{11}(0.95)^{11}(0.05)^4 - \binom{15}{12}(0.95)^{12}(0.05)^3 -$$
$$\displaystyle \binom{15}{13}(0.95)^{13}(0.05)^2 - \binom{15}{14}(0.95)^{14}(0.05)^1 - \binom{15}{15}(0.95)^{15}(0.05)^0$$

$$\displaystyle \approx 0.00061468286968$$

Apprentice123