Sure, let's solve this problem step-by-step.
### Step 1: Determine the total number of biscuits and their types.
- Total number of biscuits: 21
- Plain biscuits: 9
- Chocolate biscuits: 7
- Currant biscuits: 5
### Step 2: Calculate the probability that both biscuits are the same type.
#### Probability of drawing two plain biscuits:
- Probability of the first biscuit being plain: [tex]\(\frac{9}{21}\)[/tex]
- Probability of the second biscuit being plain (once one plain biscuit is already taken): [tex]\(\frac{8}{20}\)[/tex]
So, the probability of both being plain is:
[tex]\[ \frac{9}{21} \times \frac{8}{20} = \frac{72}{420} = \frac{36}{210} \approx 0.17142857142857143 \][/tex]
#### Probability of drawing two chocolate biscuits:
- Probability of the first biscuit being chocolate: [tex]\(\frac{7}{21}\)[/tex]
- Probability of the second biscuit being chocolate (once one chocolate biscuit is already taken): [tex]\(\frac{6}{20}\)[/tex]
So, the probability of both being chocolate is:
[tex]\[ \frac{7}{21} \times \frac{6}{20} = \frac{42}{420} = \frac{21}{210} \approx 0.09999999999999999 \][/tex]
#### Probability of drawing two currant biscuits:
- Probability of the first biscuit being currant: [tex]\(\frac{5}{21}\)[/tex]
- Probability of the second biscuit being currant (once one currant biscuit is already taken): [tex]\(\frac{4}{20}\)[/tex]
So, the probability of both being currant is:
[tex]\[ \frac{5}{21} \times \frac{4}{20} = \frac{20}{420} = \frac{10}{210} \approx 0.047619047619047616 \][/tex]
### Step 3: Calculate the total probability of both biscuits being the same type.
We sum up the probabilities of each case:
[tex]\[ 0.17142857142857143 + 0.09999999999999999 + 0.047619047619047616 = 0.31904761904761905 \][/tex]
### Step 4: Calculate the probability that the two biscuits are not the same type.
The probability of the two biscuits being not the same type is given by:
[tex]\[ 1 - \text{(probability that both are the same)} = 1 - 0.31904761904761905 \approx 0.680952380952381 \][/tex]
### Conclusion
The probability that Jane takes two biscuits of not the same type is approximately [tex]\(0.680952380952381\)[/tex].
