Nancy has a box containing only 2 orange counters and 4 white counters. She takes a counter from the box at random and does not replace it. She then takes another counter from the box at random. a) Is it more likely that Nancy takes two counters of the same colour or two of different colours? b) What is the difference between the two probabilities? Give your answer as a fraction in its simplest form.

Question

UpStudy AI · Accepted Answer

To solve the problem, we need to calculate the probabilities of Nancy taking two counters of the same color and two counters of different colors.

**Step 1: Calculate the total number of counters.**
- Orange counters: 2
- White counters: 4
- Total counters = 2 + 4 = 6

**Step 2: Calculate the probability of taking two counters of the same color.**

1. **Two orange counters:**
   - The probability of taking the first orange counter: $$ \frac{2}{6} $$
   - The probability of taking the second orange counter (after taking the first): $$ \frac{1}{5} $$
   - So, the probability of taking two orange counters is:
     $$
     P(\text{2 orange}) = \frac{2}{6} \times \frac{1}{5} = \frac{2}{30} = \frac{1}{15}
     $$

2. **Two white counters:**
   - The probability of taking the first white counter: $$ \frac{4}{6} $$
   - The probability of taking the second white counter (after taking the first): $$ \frac{3}{5} $$
   - So, the probability of taking two white counters is:
     $$
     P(\text{2 white}) = \frac{4}{6} \times \frac{3}{5} = \frac{12}{30} = \frac{2}{5}
     $$

3. **Total probability of taking two counters of the same color:**
   $$
   P(\text{same color}) = P(\text{2 orange}) + P(\text{2 white}) = \frac{1}{15} + \frac{2}{5}
   $$
   To add these fractions, we need a common denominator. The least common multiple of 15 and 5 is 15.
   $$
   P(\text{2 white}) = \frac{2}{5} = \frac{6}{15}
   $$
   Therefore,
   $$
   P(\text{same color}) = \frac{1}{15} + \frac{6}{15} = \frac{7}{15}
   $$

**Step 3: Calculate the probability of taking two counters of different colors.**

1. **Two different colors (one orange and one white):**
   - There are two scenarios: orange first, then white; or white first, then orange.

- **Scenario 1: Orange first, then white**
     - Probability of taking the first orange: $$ \frac{2}{6} $$
     - Probability of taking a white counter next: $$ \frac{4}{5} $$
     - So, the probability for this scenario is:
       $$
       P(\text{orange first, white second}) = \frac{2}{6} \times \frac{4}{5} = \frac{8}{30} = \frac{4}{15}
       $$

- **Scenario 2: White first, then orange**
     - Probability of taking the first white: $$ \frac{4}{6} $$
     - Probability of taking an orange counter next: $$ \frac{2}{5} $$
     - So, the probability for this scenario is:
       $$
       P(\text{white first, orange second}) = \frac{4}{6} \times \frac{2}{5} = \frac{8}{30} = \frac{4}{15}
       $$

2. **Total probability of taking two counters of different colors:**
   $$
   P(\text{different colors}) = P(\text{orange first, white second}) + P(\text{white first, orange second}) = \frac{4}{15} + \frac{4}{15} = \frac{8}{15}
   $$

**Step 4: Compare the probabilities.**
- Probability of same color: $$ \frac{7}{15} $$
- Probability of different colors: $$ \frac{8}{15} $$

Since $$ \frac{8}{15} > \frac{7}{15} $$, it is more likely that Nancy takes two counters of different colors.

**Step 5: Calculate the difference between the two probabilities.**
$$
\text{Difference} = P(\text{different colors}) - P(\text{same color}) = \frac{8}{15} - \frac{7}{15} = \frac{1}{15}
$$

**Final Answers:**
a) It is more likely that Nancy takes two counters of different colors.
b) The difference between the two probabilities is $$ \frac{1}{15} $$.
a) It is more likely that Nancy takes two counters of different colors. b) The difference between the two probabilities is $$ \frac{1}{15} $$.

Real Tutor Solution

Answer

Solution

Mind Expander

Related Questions

Latest Probability Questions