Probability

Overview Prob­a­bil­ity


There is some jar­gon to learn: mutu­ally exclu­sive events, com­bined events, expected fre­quency, rel­a­tive fre­quency, exper­i­men­tal prob­a­bil­ity, esti­mated prob­a­bil­ity, inde­pen­dent events and the­o­ret­i­cal prob­a­bil­ity. This may seem a lit­tle intim­i­dat­ing but with a lit­tle prac­tice their mean­ing becomes clear and easy to remem­ber. Just to irri­tate me it turns out that rel­a­tive fre­quency, exper­i­men­tal prob­a­bil­ity and esti­mated prob­a­bil­ity all the mean the same thing!

Tree dia­grams are deemed to be the most advanced sub-topic (Grade A and A*) within the Maths GCSE. I’m not sure why this is the case and if you can under­stand the basics of prob­a­bil­ity its easy to use tree diagrams.

As usual its best to get stuck into some questions.

Mutu­ally Exclu­sive Events, Expec­ta­tion, Rel­a­tive Fre­quency, Inde­pen­dent Events

Ques­tion

a) A school sells 4,800 raf­fle tick­ets. There are four dif­fer­ent colours; (T)aupe, (S)almon, (B)eige and ℗ink.

On the first draw:-

P(T) 1/3         P(S) =5 ⁄12        P(B) = 1/4

Work out the prob­a­bil­ity of pick­ing a Pink ticket on the first draw.

How many tick­ets of each colour have been sold?

b) The prob­a­bil­ity of any cus­tomer enter­ing a super­mar­ket, buy­ing a loaf of bread is 0.45. On Sat­ur­day there are are expected to be 3,200 cus­tomers, how many cus­tomers can be expected to buy a loaf of bread that day?

c) A roulette wheel has 37 slots, num­bers 1 to 36 plus an extra slot “0”. A gam­bler sus­pects that the wheel is biased towards the “0” slot. He decides to records outcomes:

Relative Frequency Question

 

 

 

 

Com­plete the table. Do you believe the wheel is biased towards the “0” slot? Explain your answer.

d) Tom and John eat a vari­ety of break­fast cere­als.  On any day there is a 40% chance that Tom will choose Corn­flakes and a 50% chance that he will choose Shred­ded Wheat. There is an 80% chance that John will choose Corn­flakes and a 10% chance he will choose Shred­ded Wheat. On any day,

  • What is the prob­a­bil­ity that both Tom and John choose Cornflakes?
  • What is the prob­a­bil­ity that nei­ther Tom nor John choose Shred­ded Wheat?
  • What is the prob­a­bil­ity that nei­ther Tom nor John choose Corn­flakes or Shred­ded Wheat?

Approach

Part (a)This is about mutu­ally exclu­sive events. These are events that can­not occur at the same time. In this case only one colour ticket can be drawn first. If, for exam­ple, a salmon ticket is drawn that excludes the pos­si­bil­ity of any other colour being drawn. You need to know that P(X) is the prob­a­bil­ity of X hap­pen­ing. Prob­a­bil­i­ties are often expressed as frac­tions so you may have to remem­ber “frac­tions, dec­i­mals and per­cent­ages”.

Part (b) This is about Expected Frequency.

Expected Fre­quency = Prob­a­bil­ity of an Event Occur­ring x Num­ber of Trials

In this exam­ple, Event Occur­ring = A cus­tomer buy­ing a loaf of bread and the num­ber of tri­als = The num­ber of cus­tomers enter­ing the supermarket.

Part © This is about Rel­a­tive Fre­quency (could be referred to as Esti­mated or Exper­i­men­tal Prob­a­bil­ity) and the­o­ret­i­cal probability.

Rel­a­tive Fre­quency = no. suc­cess­ful trials/ total no. of trials.

The­o­ret­i­cal Prob­a­bil­ity = the prob­a­bil­ity that a cer­tain out­come will occur, as deter­mined through rea­son­ing or cal­cu­la­tion. So in this ques­tion the the­o­ret­i­cal prob­a­bil­ity that the roulette wheel ball will land in any slot (assum­ing the wheel is not biased) is 1/37 as there are 37 slots (1 through 36 plus the “0” slot).

The more tri­als that are car­ried out the more you would expect the Rel­a­tive Fre­quency to approx­i­mate to the The­o­ret­i­cal Prob­a­bil­ity. For exam­ple if the tossed a coin 3 times you might not be too sur­prised to get a Rel­a­tive Fre­quency for “Heads” of 3/3 = 1.0 but if you tossed a coin a hun­dred coins you would find (assum­ing that the coin is not biased) that the Rel­a­tive Fre­quency would come close to the The­o­ret­i­cal Prob­a­bil­ity of 0.5.

Part (d) This is about inde­pen­dent events. In this ques­tion Tom’s and John’s choices are inde­pen­dent of each other. To cal­cu­late the prob­a­bil­ity of two inde­pen­dent events occur­ring you mul­ti­ply their prob­a­bil­i­ties. So if the chance of Y is 1/3 and the chance of Z is 2/3, then the prob­a­bil­ity of Y and Z occur­ring is:

P(Y) = 1/3     X     P(Z) = 2/3    = 1/3 x 2/3 = 2/9

Answer

a) The prob­a­bil­ity of a Pink ticket being drawn is

1 −(1÷3 + 5/12 + 1/4) Find the com­mon denom­i­na­tor = 1– (4÷12 + 5/12 + 3/12) = zero!

The num­ber of each colour tick­ets sold is:

Taupe = 1/3 x 4,800 = 1,600

Salmon = 5/12 x 4,800 = 2,000

Beige = 1/4 x 4,800 = 1,200

Pink none sold

Total 1,600 + 2,000 + 1,200 = 4,800 Tip– with these sort of ques­tions it’s always good to check that your cal­cu­la­tions add up to the expected total.

b) The num­ber of cus­tomers that may be expected to buy a loaf of bread is:

3,200 x 0.45 = 1,440

c) The com­pleted table is as follows:-

Relative Frequency answer

 

 

 

 

The the­o­ret­i­cal prob­a­bil­ity of the ball land­ing in the “0” slot = 1/37 = .027. You would expect that the Rel­a­tive Fre­quency would approx­i­mate to this and that it would become closer to the the­o­ret­i­cal prob­a­bil­ity with more tri­als or spins. In fact the rel­a­tive fre­quency after 1,000 spins is 0.045, which is con­sid­er­ably higher than the the­o­ret­i­cal prob­a­bil­ity. So it seems pos­si­ble that the wheel is biased to give a “0” result.

d) The prob­a­bil­ity that both Tom and John choose corn­flakes is

0.4 x 0.8 = 0.32 (32%)

The prob­a­bil­ity that nei­ther Tom nor John choose Shred­ded Wheat is

(1– 0.50) x (1– 0.1) = 0.5 x 0.9 = 0.45 (45%)

The prob­a­bil­ity that nei­ther Tom nor John will choose Corn­flakes or Shred­ded Wheat is:

(1 — 0.4 — 0.5) x (1– 0.8 — 0.1) = 0.1 x 0.1 = 0.01 (1%)

Tree Dia­gram Ques­tion and Answer to fol­low.…

 

 

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