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 | | | 2 | 36 | (unknown) | | 3 | 30 | | A coin is chosen at random from all 90 coins. The probability that the chosen coin is gold is . Given that the chosen coin is gold, which choice gives the probability that it came from Box 2?

Solution available with step-by-step explanation

SAT Probability Question 51 | Free SAT Prep | Aniko

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Question 51·Hard·Probability and Conditional Probability

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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?

When the coin is known to be gold, think: “Of all gold coins, what fraction are from Box 2?” If the coin is chosen uniformly from a known total, convert probabilities/fractions into actual counts (total gold, then gold per box). Finally compute $\dfrac{\text{gold in Box 2}}{\text{total gold}}$ and simplify.

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Use the overall probability to find a total

Because a coin is chosen from all 90 coins, $P(\text{gold})=\dfrac{\#\text{gold}}{90}$ . Use $\frac{7}{15}$ to find how many gold coins there are in total.

Convert the fractions in Boxes 1 and 3 into gold-coin counts

Multiply each box’s total coins by its gold fraction to get the number of gold coins in that box (for Box 1 and Box 3).

Find Box 2’s gold coins by subtraction

Total gold coins = (gold in Box 1) + (gold in Box 2) + (gold in Box 3). Solve for the missing amount in Box 2.

Finish with “fraction of all gold that is from Box 2”

Once you know how many gold coins are in Box 2 and how many gold coins exist total, take $\dfrac{\text{gold in Box 2}}{\text{total gold}}$ .

Desmos Guide

Compute the total number of gold coins

In Desmos, compute 90*(7/15) to get the total number of gold coins among all 90 coins.

Compute the number of gold coins in Boxes 1 and 3

Compute 24*(5/8) for Box 1 gold coins and 30*(2/5) for Box 3 gold coins.

Compute Box 2 gold coins and form the conditional probability

Compute (90*(7/15)) - (24*(5/8)) - (30*(2/5)) for Box 2 gold coins. Then divide that by 90*(7/15) to get the fraction of all gold coins that are from Box 2; match it to the answer choices.

Step-by-step Explanation

Turn the overall gold probability into a total gold count

Since every one of the 90 coins is equally likely to be chosen,

P(\text{gold})=\frac{\#\text{gold coins}}{90}.

Given $P(\text{gold})=\frac{7}{15}$ , the total number of gold coins is

90\cdot \frac{7}{15}.

Find how many gold coins are in Boxes 1 and 3

Box 1 has 24 coins and $\frac{5}{8}$ are gold:

24\cdot \frac{5}{8}.

Box 3 has 30 coins and $\frac{2}{5}$ are gold:

30\cdot \frac{2}{5}.

Use subtraction to find how many gold coins are in Box 2

The gold coins in Box 2 equal:

(\text{total gold coins})-(\text{gold in Box 1})-(\text{gold in Box 3}).

Of all gold coins, take the fraction that come from Box 2

First compute the totals:

\text{total gold} = 90\cdot\frac{7}{15}=42,

\text{gold in Box 1} = 24\cdot\frac{5}{8}=15,\qquad \text{gold in Box 3} = 30\cdot\frac{2}{5}=12.

So gold in Box 2 is

42-15-12=15.

Given that the coin is gold, the probability it came from Box 2 is the fraction of all gold coins that are in Box 2:

\frac{15}{42}=\frac{5}{14}.

Therefore, the correct choice is $\frac{5}{14}$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation