Question

1. There are 5 white and 45 black marbles in an urn. Standing next to the urn, you close your eyes and draw 10 marbles with replacement.

a. What is the probability that you draw exactly 4 white marbles?

b. What is the probability that you draw at least 6 white marbles?

2. Studies show that gasoline use for compact cars sold in the US is normally distributed with a mean of 25.5 miles per gallon (mpg) and a standard deviation of 4.5 mpg. What percentage of compacts get 30 mpg or more?

Answer 1

1) let X : be the event of drawing white marbles ( with replacement) which follows binomial distribution !

  Binomial distribution: P(X=x) = ⁿCx p^x q^n-x

P(white marble) = 5/(5+45) = 0.1 => p= 0.1 and q=1- p = 0.9 and n=10

a)  probability of exactly 4 white marbles = P(X=4)= ¹⁰C₄ * (0.1)⁴ * (0.9)^10-4 = 0.011160 [answer]

b) P(at least 6 white marbles) = P(X= 6) + P(X= 7) + P(X= 8) + P(X= 9) + P(X=10)

= ¹⁰C₆ *(0.1)⁶ * (0.9)⁴ + ¹⁰C₇ *(0.1)⁷ * (0.9)³ + ¹⁰C₈ *(0.1)⁸ * (0.9)² + ¹⁰C₉ * (0.1)⁹ *(0.9) + ¹⁰C₁₀ * (0.1)¹⁰ *(0.9)⁰

= 0.000147 [ answer]

2)  

Given that ,

mean = = 25.5

standard deviation = = 4.5

P(x 30) = 1 - P(x   30)

= 1 - P[(x - ) / (30 - 25.5) / 4.5]

= 1 -  P(z 1)  

= 1 - 0.8413

= 0.1587

Answer = 15.87%

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