PROBABILITY EXAMPLE 3
PROBLEM
I wonder how likely it is that I draw out 3 red cards from a pack of 14 cards? I think it is unlikely, perhaps I might get it once or twice. [If there were 10 red cards in your pack or even more, would you really be that surprised at drawing 3 red cards from it? This means you have to be a bit careful when you make this prediction at Merit and Excellence level .]
PLAN
I have 14 cards in my pack, 6 red and 8 black [Again, this is the similar to Example 1 but there are 14 cards instead of 7 (twice as many)]. I will shuffle the cards face down and turn up three cards at random. I will record the results in a table. I will then return the cards to the pack and reshuffle them. I will repeat this process 50 times. I will use the data to draw a bar chart. The possible outcomes are:
0 red cards
1 red card
2 red cards
3 red cards
0 red cards
1 red card
2 red cards
3 red cards
DATA
ANALYSIS
I notice that the most frequent result was getting 1 red card which was 36% of the time followed by 2 red cards 32% of the time. I got no red cards 22% of the time and 3 red cards 10% of the time. The graph is almost symmetrical but has slight skewing to the right because getting no red cards and 1 red card happen more often than getting 2 red cards and 3 red cards.
I got 3 red cards after my third trial which explains why I have a large swing in the long run frequency graph, but it gradually settles down to probability of 0.1 or 10% after 50 trials.
THEORETICAL RESULTS
You might like to read the section on probability trees first before continuing here. The concept of NOT RED is used here. That is, when a red card is not drawn, use the probability of a black card being drawn. Not a RED card = 8/14. [You only need the calculate the probability of getting 3 reds in your experiment, the other probabilities are shown here for interest.]
The probability of getting 3 red cards at random from the pack is 6/14 x 5/13 x 4/12 = 0.055
The probability of getting 2 red cards (2 red and one not red) at random from the pack is
3(6/14 x 5/13 x 8/12) = 0.33
The probability of getting 1 red card (1 red and 2 not red) at random from the pack is
3(6/14 x 8/13 x 7/12) = 0.46
The probability of getting 0 red cards at random from the pack is 8/14 x 7/13 x 6/12 = 0.15
The probability of getting 3 red cards at random from the pack is 6/14 x 5/13 x 4/12 = 0.055
The probability of getting 2 red cards (2 red and one not red) at random from the pack is
3(6/14 x 5/13 x 8/12) = 0.33
The probability of getting 1 red card (1 red and 2 not red) at random from the pack is
3(6/14 x 8/13 x 7/12) = 0.46
The probability of getting 0 red cards at random from the pack is 8/14 x 7/13 x 6/12 = 0.15
CONCLUSION
My experiment shows me I am more likely to get 1 red card as an outcome than any other. Out of 50 trials this happened 36% of the time. However, choosing 2 red cards happened 32% of the time, only 4% less that getting 1 card. No red cards was next on 22% and least likely was getting 3 red cards on 10%. This surprised me as I didn't expect to get 3 red cards happening so much.
According to theoretical probability the chance of getting 3 red cards is 0.05. It should happen 5% of the time. However, I have only conducted 50 trials. This is quite a small sample size. If I repeated this experiment again with 50 trials I expect I would get similar results. I also suspect that if I conducted much larger trials, in the long run, say 1000, my results would approach the theoretical value of 0.05. Another reason why I didn't get a result close to the theoretical result might be the way I shuffle the cards on my desk. Perhaps I didn't shuffle them enough and that's why I selected more red cards than expected. Professional lotteries use rotating barrels where the cards (or balls like in lotto) are rotated or spun for a while before being randomly selected.
According to theoretical probability the chance of getting 3 red cards is 0.05. It should happen 5% of the time. However, I have only conducted 50 trials. This is quite a small sample size. If I repeated this experiment again with 50 trials I expect I would get similar results. I also suspect that if I conducted much larger trials, in the long run, say 1000, my results would approach the theoretical value of 0.05. Another reason why I didn't get a result close to the theoretical result might be the way I shuffle the cards on my desk. Perhaps I didn't shuffle them enough and that's why I selected more red cards than expected. Professional lotteries use rotating barrels where the cards (or balls like in lotto) are rotated or spun for a while before being randomly selected.