50 people were asked if they speak French or German or Spanish. Of these people, 31 speak French 2 speak French, German and Spanish 4 speak French and Spanish but not German 7 speak German and Spanish 8 do not speak any of the languages all 10 people who speak German speak at least one other language Two of the 50 people are chosen at random. Work out the probability that they both only speak Spanish.

GCSE Edexcel Maths Geometry Toolkit: 50 people were asked if they spe

50 people were asked if they speak French or German or Spanish - Edexcel - GCSE Maths - Question 21 - 2018 - Paper 3

Question 21

50 people were asked if they speak French or German or Spanish. Of these people, 31 speak French 2 speak French, German and Spanish 4 speak French and Spa... show full transcript

Worked Solution & Example Answer:50 people were asked if they speak French or German or Spanish - Edexcel - GCSE Maths - Question 21 - 2018 - Paper 3

Step 1

Calculate the number of people who speak only Spanish

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Answer

To find the number of people who speak only Spanish, we should first determine how many people speak Spanish in total. From the information given:

Number of people who speak French and Spanish but not German: 4
Number of people who speak German and Spanish: 7
Number of people who speak French, German and Spanish: 2

Thus, the total number of Spanish speakers can be calculated as:

$ext{Total Spanish Speakers} = (4 + 2 + 7) = 13$

We already know that there are 10 people who speak German, and since all of them speak at least one other language, and 8 speak none of the languages, we can conclude the remaining must be the French speakers. Thus, there are people who only speak Spanish as follows:

$\text{Only Spanish} = 13 - (\text{Those speaking any language}) = 13 - 8 = 5$

Step 2

Calculate the probability that both chosen speak only Spanish

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Answer

To find the probability that both randomly chosen individuals speak only Spanish, we use the formula for probability:

$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$

Where the total outcomes for the selection of 2 individuals from 50 are:

$\text{Total outcomes} = \binom{50}{2} = \frac{50 \times 49}{2} = 1225$

The number of favorable outcomes (choosing 2 from those who only speak Spanish) is:

$\text{Favorable outcomes} = \binom{5}{2} = \frac{5 \times 4}{2} = 10$

Thus, the probability that both individuals selected speak only Spanish is:

$P(Both Speak Only Spanish) = \frac{10}{1225} = \frac{2}{245}$

50 people were asked if they speak French or German or Spanish - Edexcel - GCSE Maths - Question 21 - 2018 - Paper 3

Worked Solution & Example Answer:50 people were asked if they speak French or German or Spanish - Edexcel - GCSE Maths - Question 21 - 2018 - Paper 3

Calculate the number of people who speak only Spanish

Calculate the probability that both chosen speak only Spanish

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