Given that D is the midpoint of AB and B is the midpoint of AC, which statement must be true?
1 answer:
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Answer:
- 11/68 or 0.1618
Step-by-step explanation:
<u>Balls</u>
- Black - 6
- White - n
<u>Options of getting 3 balls</u>
- 1. White, black, black
- 2. Black, white, black
- 3. Black, black, white
Probability P(n) in each option
<u>1. WBB</u>
- P(w) = 6/(n + 6)
- P(b1) = n/(n + 6 - 1) = n/(n + 5)
- P(b2) = (n - 1)/(n + 5 - 1) = (n - 1)/(n + 1)
P(n) =
- P(w)P(b1)(P(b2) =
- 6/(n+6) × n/(n + 5) × (n - 1)/(n + 4) =
- 6n(n - 1)/(n + 6)(n + 5)(n + 4)
<u>2. BWB</u>
- P(b1) = n/(n + 6)
- P(w) = 6/(n + 6 - 1) = 6/(n + 5)
- P(b2) = (n - 1)/(n + 5 - 1) = (n - 1)/(n + 4)
P(n) =
- P(b1)P(w)(P(b2) =
- n/(n+6) × 6/(n + 5) × (n - 1)/(n + 4) =
- 6n(n - 1)/(n + 6)(n + 5)(n + 4)
<u>3. BBW</u>
- P(b1) = n/(n + 6)
- P(b2) = (n - 1)/(n + 6 - 1) = (n - 1)/(n + 5)
- P(w) = 6/(n + 5 - 1) = 6/(n + 4)
P(n) =
- P(b1)P(b2)(P(w) =
- n/(n+6) × (n - 1)/(n + 5) × 6/(n + 4) =
- 6n(n - 1)/(n + 6)(n + 5)(n + 4)
<u>Final equation is same for each case:</u>
- P(n) = 6n(n - 1) / (n + 6)(n + 5)(n + 4)
The easy way to find the maximum is to try the numbers or graph.
Both of the methods give the maximum integer n = 11 or n = 12
<em>See attached graph</em>
<u>At both values n we get P(n):</u>
- P(11) = 6*11*10 / 15*16*17 = 11/68 = 0.1618 (rounded)
- P(12) = 6*12*11 / 16*17*18 = 11/68 = 0.1618 (rounded)
