Probability: part 1

Probability models

1. A bag contains 6 red, 3 blue, 2 green and 1 white balls. A ball is picked at random. What is the probablity that it is:
   1. red
   2. blue or white
   3. not green (hint: think 'complement')
   4. not green or red?
2. A card is selected randomly from a pack of 52. What is the probability that it is:
   1. the 2 of hearts
   2. a red card
   3. a picture card
   4. an ace
   5. a number less than 4?
3. Even numbers from 2 -100 are written on cards. What is the probability of selecting a multiple of 5, if a card is drawn at random?
   

Probability identities

The following results apply to probabilities, for the sample space $S$ and two events $A$ and $B$ , within $S$ .

We can demonstrate this last result using a Venn diagram. The union of A and B is the set of all elements in A or in B or in both.

The probability of event A occurring is given by $\mathrm{P(A)}$ and the probability of event B occurring is given by $\mathrm{P(B)}$ . However, if we look closely at the circle representing either of these events, we notice that the probability includes a small part of the other event. So event A includes a bit of event B and vice versa. This is shown in the following figure:

So to find the probability of $P(A\cup B)$ we notice the following:

• We can add $P\left(A\right)$ and $P\left(B\right)$
• Doing this counts the intersection twice, once in $P\left(A\right)$ and once in $P\left(B\right)$ .

The following video provides a brief summary of some of the work covered so far.

Probability identities

Answer the following questions

1. Rory is target shooting. His probability of hitting the target is $\mathrm{0,7}$ . He fires five shots. What is the probability that all five shots miss the center?
   
2. An archer is shooting arrows at a bullseye. The probability that an arrow hits the bullseye is $\mathrm{0,4}$ . If she fires three arrows, what is the probability that all the arrows hit the bullseye?
   
3. A dice with the numbers 1,3,5,7,9,11 on it is rolled. Also a fair coin is tossed. What is the probability that:
   1. A tail is tossed and a 9 rolled?
   2. A head is tossed and a 3 rolled?
4. Four children take a test. The probability of each one passing is as follows. Sarah: $\mathrm{0,8}$ , Kosma: $\mathrm{0,5}$ , Heather: $\mathrm{0,6}$ , Wendy: $\mathrm{0,9}$ . What is the probability that:
   1. all four pass?
   2. all four fail?
5. With a single pick from a pack of 52 cards what is the probability that the card will be an ace or a black card?
   

Mutually exclusive events

Two events are called mutually exclusive if they cannot be true at the same time.

Examples of mutually exclusive events are:

1. A die landing on an even number or landing on an odd number.
2. A student passing or failing an exam
3. A tossed coin landing on heads or landing on tails

This means that if we examine the elements of the sets that make up $A$ and $B$ there will be no elements in common. Therefore, $A\cap B=\varnothing$ (where $\varnothing$ refers to the empty set). Since, $P(A\cap B)=0$ , equation [link] becomes:

for mutually exclusive events.

We can represent mutually exclusive events on a Venn diagram. In this case, the two circles do not touch each other, but are instead completely separate parts of the sample space.

Mutually exclusive events

1. A box contains coloured blocks. The number of each colour is given in the following table.

   Colour	Purple	Orange	White	Pink
   Number of blocks	24	32	41	19

   A block is selected randomly. What is the probability that the block will be:
   1. purple
   2. purple or white
   3. pink and orange
   4. not orange?
2. A small private school has a class with children of various ages. The table gies the number of pupils of each age in the class.

   3 years female	3 years male	4 years female	4 years male	5 years female	5 years male
   6	2	5	7	4	6

   If a pupil is selceted at random what is the probability that the pupil will be:
   1. a female
   2. a 4 year old male
   3. aged 3 or 4
   4. aged 3 and 4
   5. not 5
   6. either 3 or female?
3. Fiona has 85 labeled discs, which are numbered from 1 to 85. If a disc is selected at random what is the probability that the discnumber:
   1. ends with 5
   2. can be multiplied by 3
   3. can be multiplied by 6
   4. is number 65
   5. is not a multiple of 5
   6. is a multiple of 4 or 3
   7. is a multiple of 2 and 6
   8. is number 1?

Random experiments

1. Let $S$ denote the set of whole numbers from 1 to 16, $X$ denote the set of even numbers from 1 to 16 and $Y$ denote the set of prime numbers from 1 to 16
   1. Draw a Venn diagram accurately depicting $S$ , $X$ and $Y$ .
   2. Find $n\left(S\right)$ , $n\left(X\right)$ , $n\left(Y\right)$ , $n(X\cup Y)$ , $n(X\cap Y)$ .
2. There are 79 Grade 10 learners at school. All of these take either Maths, Geography or History. The number who take Geography is41, those who take History is 36, and 30 take Maths. The number who take Maths and History is 16; the number who take Geography and History is 6, and there are8 who take Maths only and 16 who take only History.
   1. Draw a Venn diagram to illustrate all this information.
   2. How many learners take Maths and Geography but not History?
   3. How many learners take Geography only?
   4. How many learners take all three subjects?
3. Pieces of paper labelled with the numbers 1 to 12 are placed in a box and the box is shaken. One piece of paper is taken out andthen replaced.
   1. What is the sample space, $S$ ?
   2. Write down the set $A$ , representing the event of taking a piece of paper labelled with a factor of 12.
   3. Write down the set $B$ , representing the event of taking a piece of paper labelled with a prime number.
   4. Represent $A$ , $B$ and $S$ by means of a Venn diagram.
   5. Find
      1. $n\left(S\right)$
      2. $n\left(A\right)$
      3. $n\left(B\right)$
      4. $n(A\cap B)$
      5. $n(A\cup B)$
   6. Is $n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$ ?