Math Snap

STEP 1

What is this asking?
Out of 15 people, what's the chance that exactly 4 of them work out daily, if we know 18% of people generally do?
Also, we need to find some probabilities using a z-table or calculator.
Watch out!
Don't mix up the binomial formula parts, and make sure to use the correct z-scores for the normal distribution questions!

STEP 2

1. Binomial Probability
2. Z-score Less Than
3. Z-score Greater Than
4. Z-score Between Two Values (1)
5. Z-score Between Two Values (2)

STEP 3

The binomial probability formula tells us the chance of getting exactly $k$ successes in $n$ trials:
$$P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{(n-k)}$$ Where $p$ is the probability of success on a single trial.

STEP 4

We have $n = \textbf{15}$ people (trials), we want $k = \textbf{4}$ people to work out (successes), and the probability of someone working out daily is $p = \textbf{0.18}$.
Let's plug these into our formula!
$$P(X=4) = \binom{15}{4} \cdot (0.18)^4 \cdot (1-0.18)^{(15-4)}$$

STEP 5

The binomial coefficient $\binom{15}{4}$ represents the number of ways to choose 4 people out of 15.
It's calculated as:
$$\binom{15}{4} = \frac{15!}{4! \cdot (15-4)!} = \frac{15!}{4! \cdot 11!} = \frac{15 \cdot 14 \cdot 13 \cdot 12}{4 \cdot 3 \cdot 2 \cdot 1} = \textbf{1365}$$

STEP 6

Now, let's plug everything back into our binomial formula:
$$P(X=4) = 1365 \cdot (0.18)^4 \cdot (0.82)^{11} \approx 1365 \cdot 0.00104976 \cdot 0.1165609 \approx \textbf{0.167}$$ So, there's about a 16.7% chance of exactly 4 out of 15 people working out daily.

STEP 7

We want to find $P(z < 0.47)$.
Using a z-table or calculator, we find this probability to be approximately $\textbf{0.6808}$.

STEP 8

We want $P(z > -1.83)$.
This is the same as $1 - P(z < -1.83)$.
Using a z-table or calculator, $P(z < -1.83) \approx 0.0336$, so $P(z > -1.83) \approx 1 - 0.0336 = \textbf{0.9664}$.

STEP 9

We're looking for $P(-1.29 < z < 0.64)$.
This is equal to $P(z < 0.64) - P(z < -1.29)$.
Using a z-table or calculator, $P(z < 0.64) \approx 0.7389$ and $P(z < -1.29) \approx 0.0985$.
So, $P(-1.29 < z < 0.64) \approx 0.7389 - 0.0985 = \textbf{0.6404}$.

STEP 10

We want $P(-1.96 < z < 1.96)$.
This is $P(z < 1.96) - P(z < -1.96)$.
From a z-table or calculator, $P(z < 1.96) \approx 0.9750$ and $P(z < -1.96) \approx 0.0250$.
Therefore, $P(-1.96 < z < 1.96) \approx 0.9750 - 0.0250 = \textbf{0.95}$.

The probability of exactly 4 out of 15 people working out daily is approximately $\textbf{0.167}$.
For the standard normal distribution: a) $P(z < 0.47) \approx \textbf{0.6808}$, b) $P(z > -1.83) \approx \textbf{0.9664}$, c) $P(-1.29 < z < 0.64) \approx \textbf{0.6404}$, and d) $P(-1.96 < z < 1.96) \approx \textbf{0.95}$.