Question

extra credit
a mathematician sets up an experiment. she puts 5 pennies, 4 nickles, 7 dimes,
and 3 quarters into a box. when someone draws a coin, they can replace it at draw
again to try and get more money. however, if they replace their coin, another coin
of that same value gets added (so if you draw a penny, two pennies get put back
in). what is the probability that someone draws a nickel, wants to replace and try
again, and draws a quarter?
p(drawing a nickel first):
p(drawing a quarter second):
p(drawing a nickel, then a quarter):
independent or dependent events?

Explanation:

Step1: Calculate total initial coins

Total coins = 5 + 4 + 7 + 3 = 19

Step2: Probability of first nickel

P(nickel first) = $\frac{\text{Number of nickels}}{\text{Total initial coins}} = \frac{4}{19}$

Step3: New total coins after replacement

After drawing nickel, add 1 nickel: new nickels = 4 + 1 = 5; new total = 19 + 1 = 20

Step4: Probability of second quarter

P(quarter second) = $\frac{\text{Number of quarters}}{\text{New total coins}} = \frac{3}{20}$

Step5: Combined probability

Multiply the two probabilities: $\frac{4}{19} \times \frac{3}{20}$

Step6: Simplify the product

$\frac{12}{380} = \frac{3}{95}$

Step7: Determine event dependence

First draw changes total coins, so events are dependent.

Answer:

P(drawing a nickel first): $\frac{4}{19}$
P(drawing a quarter second): $\frac{3}{20}$
P(drawing a nickel, then a quarter): $\frac{3}{95}$
Independent or Dependent events? Dependent