What is the probability of picking 2 non face cards out of a deck of playing cards if you do not replace the first card after you draw it?

Choose an appropriate response from the probability line above for the following events: Some of the events might fall between the probabilities e.g. very unlikely or almost certain. Some responses might depend your own circumstances.1) The sun will not rise tomorrow2) You will return to your current school next year3) It will rain tomorrow4) Mozart will perform in assembly5) You will get a lunchtime detention 

6) You win Lotto

Theoretical Probability =  the number of times something is wanted 
                                             the number of possible outcomes

For example, find the probability of obtaining Heads from a coin flip. There is only one head on a coin and there are two possible outcomes, either Heads or Tails. Therefore, the probability is:

                           Probability (p) = 1      =     1/2 or 0.5 or 50%

                          Probability (p) = 1    = 1/6 or 0.17 or 17%

The symbol p or (p) is used as a short hand way of writing 'probability'.

Sometimes the activities we do are practical. Mathematicians often use a variety of tools to help them learn and understand how probability works. The whole point is to have something that can simulate a random event. Some of these tools are:1) Dice (or die  - singular). they come in a variety of shapes and sizes, not just 6 sides!2) Playing cards - very popular as they can be combined with games.3) Spinners (wheels with markings on them) another random tool4) Calculators and computer random number generators - useful but more costly

5) Software packages like Microsoft Excel and online apps like Random.Org

Fig.1 Standard deck of 52 playing cards

Teaching probability using playing cards is very popular and yet many students do not understand how they work so let's investigate that straight up.1) There are 4 suits: (Fig.1 top to bottom)     Clubs x 13     Spades x 13     Hearts x 13     Diamonds x 13

This gives a total of 52 cards.

2) Within each suit there are 13 cards: Fig.2 shows the suit of Hearts:      Ace, 2,3,4,5,6,7,8,9,10, Jack, Queen and King.3) Of the 13 cards, the Jack, Queen and King are called royal cards or face cards. The good news is that when card hands are used in mathematics the question will explain the types of cards that make up the hand.

Fig.2 "The Jack of hearts he stole those tarts and with them ran away..."

Fig.3 Probability with replacement - "put it back"

'With Replacement' means you put the balls back into the box so that the number of balls to choose from is the same for any draws when removing more than 1 ball.

Above are 10 coloured balls in a box, 4 red, 3 green, 2 blue and 1 black.  A ball is randomly selected. After each selection the balls will be returned to the box.What is the probability that if a ball or balls are randomly selected that we choose:1) a red ball?2) a black ball?3) not a blue ball?4) a blue and green ball?

5) a red ball and 2 green balls?

Probability = total number of red =     4                          total number of balls     10 1) There are 4 red balls in the box so the probability of  picking a red ball is 4 out of 10 or 4/10 which can be expressed as 40% 2) There is only one black ball in the box so the probability is 1/10 or 10%.3) There are 8 balls that are not blue, so the probability of picking a ball that's not blue is 8/10 or 80%. This makes sense because most balls are not blue in colour. 4) The probability of picking a blue ball is 2/10 and the probability of picking a green ball is 3/10. So the probability of picking both is: 2/10 x  3/10 = 6/100=0.06  or 6% 5) The probability of picking a red ball is 4/10 and the probability of picking a green ball is 3/10 and because the ball is put back in the box, the second green is also 3/10. So the probability is:4/10 x 3/10 x 3/10 = 36/1000=0.036 or 3.6%

Fig.4 Probability without replacement first ball out "don't put it back"

Fig.5 Probability without replacement second ball out.

"Without replacement" means that you don't put the ball or balls back in the box so that the number of balls in the box gets less as each ball is removed. This changes the probabilities. Let's look at question 4 above.
What is the probability that if a ball or balls are randomly selected that we choose:
a blue and green ball?The probability of choosing the blue ball is 2/10 and the probability of choosing the green ball is 3/9 because after the first ball is taken out, there are 9 balls remaining. So the probability is:

2/10 x 3/9 = 6/90 or 1/15 = 6.7% (Compare that with replacement of 6/100 or 6%)

Fig.6 House of Cards Example using probability without replacement.

Fig.6 shows 7 cards, 3 red and 4 black. The same cards can be used to explain the probabilities of House of Cards Example 3. The numbers might be different (6 red and 8 black) but the process is the same.

Remember that the type of problem was " I wonder how likely it is that I can draw 3 red cards from a pack of 7 cards". The outcomes were, 0, 1, 2 or 3 red cards. Because it is easier to work out the probabilities of 0 and 3 red cards we will calculate those probabilities first.

The probability of drawing the first ball is 3/7 but after that there are only 2 red cards and 6 cards in total.

The probability of drawing the 2nd card is 2/6 but after that there is only 1 red card and 5 cards in total.

The probability of drawing the 3rd card is 1/5.

The probability of drawing all 3 red cards can be found by multiplying their probabilities together. This gives  3/7 x 2/6 x 1/5 = 6/210 or 3%