Question

a restaurant has determined a probability distribution for the number of steaks that it sells in a given day:

number sold (x)	0	4	8	12	16
probability p(x) = x	0.21	0.11	0.35	?	0.12

write all answers as decimals.
what is the missing probability at x = 12?
what is the probability of selling 12 steaks in a given day?
what is the probability of selling at least 8 steaks?
what is the probability of selling 4 or 12 steaks?
what is the probability of selling 20 steaks?
what is the expected number of steaks sold in a day?
what is the probability of selling at most 8 steaks?

Explanation:

Step1: Find missing probability

The sum of all probabilities in a probability - distribution is 1. Let the missing probability at $x = 12$ be $p$. Then $0.21+0.11 + 0.35+p + 0.12=1$. So, $p=1-(0.21 + 0.11+0.35 + 0.12)=1 - 0.79 = 0.21$.

Step2: Probability of selling 12 steaks

The probability of selling 12 steaks is the value of $P(X = 12)$, which is 0.21.

Step3: Probability of selling at least 8 steaks

$P(X\geq8)=P(X = 8)+P(X = 12)+P(X = 16)=0.35+0.21+0.12 = 0.68$.

Step4: Probability of selling 4 or 12 steaks

Since the events are mutually - exclusive, $P(X = 4\ or\ X = 12)=P(X = 4)+P(X = 12)=0.11+0.21 = 0.32$.

Step5: Probability of selling 20 steaks

Since 20 is not in the given probability distribution, $P(X = 20)=0$.

Step6: Calculate expected value

The expected value $E(X)=\sum_{i}x_{i}P(x_{i})=0\times0.21 + 4\times0.11+8\times0.35+12\times0.21+16\times0.12=0 + 0.44+2.8+2.52+1.92 = 7.68$.

Step7: Probability of selling at most 8 steaks

$P(X\leq8)=P(X = 0)+P(X = 4)+P(X = 8)=0.21+0.11+0.35 = 0.67$.

Answer:

Missing probability at $x = 12$: 0.21
Probability of selling 12 steaks: 0.21
Probability of selling at least 8 steaks: 0.68
Probability of selling 4 or 12 steaks: 0.32
Probability of selling 20 steaks: 0.00
Expected number of steaks sold in a day: 7.68
Probability of selling at most 8 steaks: 0.67