Compound Probability With and Without Replacement

Compound Probability With and Without Replacement exercise

• What is dependent probability?

  Hints

  The word dependent means 'the outcome of one affects the outcome of another'.

  Look for the choice above which means that.

  An example of a dependent event is a person going to work on a train and bus. If they miss the train then they miss the bus. In other words, catching the bus is dependent on catching the train first.

  Solution

  Where the first event affects the probability of the following events.

  The second event is dependent on the first one. For example, in a knockout competition, to get to the second round a team must win the first round. Participation in the second round depends on winning the first round.

• Finding dependent probability.

  Hints

  We calculate the probabilities separately first.

  Probability of picking a cola is $\frac{Cola}{Total}$

  We multiply this by the second probability when the cola has NOT been replaced.

  Probability of picking a cream soda is $\frac{Cream Soda}{Total - 1}$

  We work out dependent probabilities like this:

  Solution

  $\mathbf{\frac{12}{56}}$

  We work it out like this:

  • To pick a cola there are $3$ out of a total of $8$ cans
  • We write that as $\frac{3}{8}$
  • To pick a cream soda when the other can has NOT been replaced we have $4$ out of $(8 - 1) = 7$
  • We write that as $\frac{4}{7}$
  • Multiply the numerators ($3\times4 = 12$)
  • Multiply the denominators ($8\times7 = 56$)

• Work out the mixed probability questions.

  Hints

  Probability of picking an item is $\dfrac{\text{Number of items}}{\text{Total items}}$.

  For example if there are $3$ pink items in a bag of $7$ items, we write the probability as $\frac{3}{7}$

  We calculate two independent probabilities separately first.

  Then multiply them together.

  We calculate two dependent probabilities separately first.

  Then multiply them together.

  When we have had the first pick from the bag and the sweet is not replaced there is $1$ less sweet in the bag for the next pick.

  This is dependent probability.

  Solution

  • $1$ fruit sweet

  $\frac{2}{6} = \frac{1}{3}$

  • $1$ toffee sweet

  $\frac{4}{6} = \frac{2}{3}$

  • $1$ fruit then $1$ toffee with replacement

  $\frac{2}{6}\times\frac{4}{6} = \frac{8}{36} = \frac{2}{9}$

  • $1$ fruit then $1$ toffee without replacement

  $\frac{2}{6}\times\frac{4}{5} = \frac{8}{30} = \frac{4}{15}$

• Calculate dependent probability.

  Hints

  We calculate the probabilities separately first.

  Probability of picking a blue sock is: $\frac{\text{Blue}}{\text{Total}}$

  We multiply this by the second probability when the blue sock has NOT been replaced.

  Probability of picking another blue sock is remaining number of $\frac{\text{Blue}}{\text{Total - 1}}$

  We calculate the dependent probability of finding two blue socks like this.

  In this example, there are $3$ blue socks out of a total of $6$ socks in the drawer.

  When one has been removed there are then only $2$ blue socks over a total of $5$ socks.

  Solution

  $\mathbf{\frac{30}{72}}$.

  Probability of picking two blue socks is shown here.

  There are $6$ to start with and when $1$ is picked $5$ remain. Multiply these together.

  For the denominator there are $9$ socks to start with and when $1$ is removed there are $8$. Multiply these together.

  We can simplify the answer by dividing by $6$ to give us $\frac{5}{12}$

• Finding independent probability.

  Hints

  We calculate the probabilities separately first.

  Probability of picking an apple is $\frac{Apples}{Total}$

  We multiply this by the second probability when the apple has been replaced.

  Probability of picking a banana is $\frac{Bananas}{Total}$

  We multiply probabilities like this:

  Solution

  $\mathbf{\frac{4}{100}}$

  The apple was replaced, so there were still $10$ pieces of fruit in the bowl.

  • There are $4$ apples out of $10$ pieces of fruit in the bowl
  • We write this as $\frac{4}{10}$
  • The apple is replaced into the bowl
  • There is only $1$ banana in the bowl, so we write this as $\frac{1}{10}$
  • Multiply the numerators, $4$ apples by the $1$ banana $= 4$
  • Multiply the denominators, $10$ total pieces of fruit and $10$ total pieces of fruit $= 100$

• Finding dependent probabilities.

  Hints

  The probability is dependent.

  The second pick will have one less spanner and one less tool in the shed.

  The third pick will have two less.

  To work out the total we multiply all $3$ fractions together.

  Solution

  $\frac{4}{7}\times\frac{3}{6}\times\frac{2}{5} = \frac{24}{210}$

  $4$ spanners to start with and one less for each pick.

  Also, there are $7$ tools to start with and one less each pick.