It is believed that 15% of teenagers in England of the age of sixteen are smokers. Suppose that 10 sixteen year olds are selected at random from the population. 1. What is the probability that exactly two of the ten smoke? 2. What is the probability that three of fewer of the ten smoke? 3. What is the probability that over half of the ten are non-smokers? Please can you include working out. Thanks
Oh man, saw this a while ago @ school (couple years or so..) wasn't good at it hope someone can help you with it, good luck!
I'm going to help you do these with the assumption that you can use a graphic calculator (ex. Texas Instruments). You'll need to use binomialpdf for exact problems (#1) and binomialcdf for cumulative problems (#2, #3). I believe this is right, but I'm not 100% sure. There are other methods, this is just the one I remember at the moment. If you have a TI calculator, the indexes (bpdf, bcdf) can be found by pressing 2nd > VARS in the DISTR menu -- you'll have to scroll down to the bottom.
If this is S1 for AS maths, the you can use a calculator and use the nCr button to work out Binomial Expansions: X is the Number of smokers of age 16 X~B(10,0.15) 1.) Exactly 2 smokers P(X=2) = (10C2) x (0.15)² x (0.85)⁸ = 0.276 2.) Three or fewer smoke. You can use a Cumulative Binomial Probabilty table to work this out. Look for the Row N=10 and X=3. The look for the collumn P=0.15 This should give you the answer 0.95. So P(X≤3) = 0.95 If you dnt get the Binomial Probabilty Table, then you will have to work out what P(X=3), P(X=2), P(X=1) and P(X=0) is and add them all together... which is long... so you should use the Table. 3.) Over half the ten are non smokers. This is the same as the probabilty of half or less are smokers. so again you will need to use the Cumulative Binaomial Porbabilty Table to work this out. Go back tot he Row, N=10 but this time go to the sub-row X=5 and look the the collumn P=0.15 to find the answer of P(X≤5) = 0.9986 Hope that helps, you need to do write: X is the Number of smokers of age 16 X~B(10,0.15) To get the working out marks.
I agree with this post, because I did S1 AS Maths and I was taught to do it this way as well. The cumulative binomial probability tables should be in a Formula Book given to you. I'm pretty sure that this is a 3-4 mark questions and if you set it out like that, you should get the marks for working out as well.
