Studdy AI Logo
Math

QuestionA store has hats from 2 manufacturers in 3 colors. Find the probabilities for various selections based on the data provided.
a) P(Black or Manufacturer B) b) P(Red) c) P(Manufacturer B and Red) d) P(Manufacturer B or Red)
Use decimal or fraction rounded to three places.

Studdy Solution

STEP 1

Assumptions1. The total number of hats is114. . The number of black hats is38.
3. The number of hats from Manufacturer B is56.
4. The number of red hats is45.
5. The number of red hats from Manufacturer B is26.

STEP 2

First, we need to find the probability that the hat was Black or was from Manufacturer B. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of outcomes.The formula for the probability of either of two events A or B happening is given by(A∪B)=(A)+(B)−(A∩B)(A \cup B) =(A) +(B) -(A \cap B)

STEP 3

Now, plug in the given values for the number of black hats, the number of hats from Manufacturer B, and the total number of hats to calculate the probability.
(Black∪Manufacturer B)=38114+56114−13114(Black \cup Manufacturer\, B) = \frac{38}{114} + \frac{56}{114} - \frac{13}{114}

STEP 4

Calculate the probability.
(Black∪Manufacturer B)=38114+56114−13114=81114≈0.711(Black \cup Manufacturer\, B) = \frac{38}{114} + \frac{56}{114} - \frac{13}{114} = \frac{81}{114} \approx0.711

STEP 5

Next, we need to find the probability that the hat was Red. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of outcomes.(Red)=Number of Red HatsTotal Number of Hats(Red) = \frac{Number\, of\, Red\, Hats}{Total\, Number\, of\, Hats}

STEP 6

Now, plug in the given values for the number of red hats and the total number of hats to calculate the probability.
(Red)=45114(Red) = \frac{45}{114}

STEP 7

Calculate the probability.
(Red)=45114≈0.395(Red) = \frac{45}{114} \approx0.395

STEP 8

Next, we need to find the probability that the hat was from Manufacturer B and was Red. The probability of two independent events A and B both happening is given by(A∩B)=Number of Red Hats from Manufacturer BTotal Number of Hats(A \cap B) = \frac{Number\, of\, Red\, Hats\, from\, Manufacturer\, B}{Total\, Number\, of\, Hats}

STEP 9

Now, plug in the given values for the number of red hats from Manufacturer B and the total number of hats to calculate the probability.
(Manufacturer B∩Red)=26114(Manufacturer\, B \cap Red) = \frac{26}{114}

STEP 10

Calculate the probability.
(Manufacturer B∩Red)=26114≈0.228(Manufacturer\, B \cap Red) = \frac{26}{114} \approx0.228

STEP 11

Finally, we need to find the probability that the hat was from Manufacturer B or was Red. The formula for the probability of either of two events A or B happening is given by(A∪B)=(A)+(B)−(A∩B)(A \cup B) =(A) +(B) -(A \cap B)

STEP 12

Now, plug in the given values for the number of hats from Manufacturer B, the number of red hats, the number of red hats from Manufacturer B, and the total number of hats to calculate the probability.
(Manufacturer B∪Red)=56114+45114−26114(Manufacturer\, B \cup Red) = \frac{56}{114} + \frac{45}{114} - \frac{26}{114}

STEP 13

Calculate the probability.
(Manufacturer B∪Red)=56114+45114−26114=75114≈0.658(Manufacturer\, B \cup Red) = \frac{56}{114} + \frac{45}{114} - \frac{26}{114} = \frac{75}{114} \approx0.658a) The probability that the hat was Black or was from Manufacturer B is approximately0.711. b) The probability that the hat was Red is approximately0.395. c) The probability that the hat was from Manufacturer B and was Red is approximately0.228. d) The probability that the hat was from Manufacturer B or was Red is approximately0.658.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord