Maths probability?

year10 group year 11 group


boys 75 50


girls 60 30





above, shows the number of boys and the number of girls in year10 and year11 of a school.


The headteacher wants to find out what pupils think about a new year 11 common room. A stratified sample of size 50 is to be taken from year 10 and year 11.





a) calculate the number of pupils to be sampled from year 10.





two pupils are to be chosen at randon to speak to the headteacher.





one pupil is to be chosen from year 10.


one pupil is to be chosen from year 11.





b) calculate the probability that both pupils will be boys.





however, the headteacher decides to choose one boy at random from all the boys in year 10 and 11 together and one girl at random from all the girls in year 10 and 11 together.





c) calculate the probability that exactly one of the pupils will be from year 10.

Maths probability?
There are 135 pupils in year 10 (Y10) and 80 pupils in year 11 (Y11). This makes 215 pupils in total. So 135 of the 215 pupils are in Y10, this is 62,79% (135/215 x 100%). The rest, 37,21% of the pupils, are in Y11. If you are taking a stratified sample, these ratios have to be respected.





a) calculate the number of pupils to be sampled from year 10:





A sample of 50 pupils is taken, 62,79% of all pupils are in Y10. So 62,79% of the pupils in the sample should also come from Y10. 62,79% of 50 is 31,395. So 31 pupils should be sampled from Y10.





b) calculate the probability that both pupils will be boys:





In Y10, 55,5% of the pupils are boys. So the chance to choose a boy at random from Y10 is 55,5%.


In Y11, 62,5% of the pupils are boys. So the chance to choose a boy at random from Y11 is 62,5%.


So the chance to choose a boy from Y10 AND a boy form Y11 is 55,5% x 62,5% = 34,68%





c) calculate the probability that exactly one of the pupils will be from Y10:





This probability is the sum of the probability of the two possible outcomes that matches the restraint. Either you can have a boy from Y10 and a girl from Y11 or you can have a girl from Y10 and a boy from Y11.





We begin with the probability to choose a boy from Y10 and a girl from Y11. There are 75 boys in Y10 and 50 boys in Y11. So the probability to choose a boy from Y10 is 60% (75/125 x 100%). There are 60 girls in Y10 and 30 girls in Y11. So the chance to choose a girl from Y11 is 33% (30/90 x 100%). So the probability to choose a boy from Y10 AND a girl from Y11 is 20% (60% x 33%).





Now we calculate the probability to pick a girl from Y10 and a boy from Y11. The probability to choose a girl from Y10 is 66,6% (60/90 x 100%). The probability to choose a boy from Y11 is 40% (50/125 x 100%). So the probability to choose a girl from Y10 AND a boy from Y11 is 26,66% (66,6% x 40%).





So the probability that exactly one of the pupils will be from Y10 is 46,66% (20% + 26,66%).