Question 1040·Hard·Math

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Three boxes contain a total of 90 coins. The information below is known:

Box	Number of coins in the box	Fraction of coins in the box that are gold
1	24	$\frac{5}{8}$
2	36	(unknown)
3	30	$\frac{2}{5}$

A coin is chosen at random from all 90 coins. The probability that the chosen coin is gold is $\frac{7}{15}$ . Given that the chosen coin is gold, which choice gives the probability that it came from Box 2?

The phrase given that signals a conditional probability: only outcomes meeting that condition belong in the final denominator. First turn the overall gold probability into a count, then use the box fractions to find how many gold coins are in the unknown box. Desmos can handle the counts and division. Once gold is known, divide by all gold coins, not all coins.

Hints

Hint 1
The overall gold probability describes a fraction of all the coins, not a fraction of one box. Multiply it by the total number of coins to find how many gold coins are in the three boxes together.
Hint 2
A box’s gold fraction applies to that box alone. Multiply the coin counts for Boxes 1 and 3 by their respective gold fractions. How can those two gold counts help you find the missing one?
Hint 3
In conditional probability, given that the coin is gold removes every nongold coin from the possible picks. Which total should go beneath the count of gold coins from Box 2?

Step-by-step

Count the gold coins, then restrict the group
Step 1
Find the total number of gold coins
The given $\frac{7}{15}$ is the fraction of all $90$ coins that are gold, so multiply the total by it. Type $90\left(\frac{7}{15}\right)$ ; Desmos shows $42$ gold coins.
Step 2
Count the gold coins in Box 1
Box 1 has $24$ coins, and $\frac{5}{8}$ of that box is gold. Type $24\left(\frac{5}{8}\right)$ ; Desmos shows $15$ gold coins in Box 1.
Step 3
Count the gold coins in Box 3
Box 3 has $30$ coins, and $\frac{2}{5}$ of that box is gold. Type $30\left(\frac{2}{5}\right)$ ; Desmos shows $12$ gold coins in Box 3.
Step 4
Find the missing gold count
The $42$ gold coins are spread across all three boxes, so subtract the two known gold counts to find Box 2’s gold coins. Type $42-15-12$ ; Desmos shows $15$ .
Step 5
Divide within the gold coins
The words given that the chosen coin is gold mean only gold coins remain possible. In a conditional probability, that known condition sets the denominator: $42$ gold coins, not all $90$ coins. Of those $42$ , $15$ came from Box 2. Type $15/42$ ; Desmos shows $0.357142\ldots$ . Tap the fraction button to see $\frac{5}{14}$ . So the probability that a gold coin came from Box 2 is $\frac{5}{14}$ . Choice D.

Question 1040·Hard·Math

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