Theoretical and Observed Two-category Discrete Probability Distributions Without Replacement, Statistical Note 32

Draw 20 cards without replacement from  a deck of 52 cards and count the number of black cards. Repeat the same process for seven times or sets each constituting 20 cards. Calculate the theoretical and observed discrete probability distributions of number of black cards in 20 cards.

Theoretical probability distribution gives an idea of an ideal probability distribution, what a distribution should be given the parameters. The observed probability distribution is based on the real-time data and shows how different the distribution is from the ideal situation. Sampling distribution helps compare the theoretical and observed distributions.

Drawing some cards without replacement from a deck of 52 cards is an example of the two-category discrete probability distribution of sampling without replacement. Refer to my earlier Statistical Notes for clarity on calculating the two-category discrete probability using tree diagram, formula and Excel software function.

  

Theoretical Discrete Probability Distribution

I discussed on the Theoretical Two-Category Discrete Probability Distribution of sampling with replacement in my former Statistical Note 31. Here, I present only the table constituting the number of black cards in 20 cards drawn without replacement from a deck of cards and respective probabilities (Table 1). 

Table 1: Number of Black Cards in 20 Cards Drawn Without Replacement from a Deck of Cards and Respective Probabilities

Occurrence of 10 black cards in 20 cards has the highest probability (highlighted yellow) and is thus, highly likely to occur. The likelihood decreases towards both sides of 10 black cards. Two extreme number of black cards, 0 and 20, have the least chance of occurrence.

Trial Data

I drew 20 cards without replacement from a deck of 52 cards in a set and the same process was repeated for seven sets or times. Table 2 presents the outcome of 20 cards drawn without replacement in each of seven sets. Black and red cards were coded one and zero respectively for symbolic representation.

Table 2: Outcomes in 20 cards drawn without replacement from a deck of cards in each of seven sets (Black card=1 and Red card=0)

To summarize, the number of black cards in seven sets ranged from six to seven, nine and then 11 to 13 (Table 2). This is due to the sampling error. The observed mean number of black cards is the sum of the number of black cards from each of seven sets divided by seven, equal to 10.

Observed Discrete Probability Distribution

In further summary, it is noted that 12 black cards occurred twice in two of seven sets of 20 cards (Table 3). Thus, occurrence of 12 black cards is most likely to occur with the probability P(X=12)=0.285714.  Other five samples had non-repetitive number of black cards that occurred in 20 cards in each sample.

Table 3: Number of black cards in 20 cards drawn without Replacement in each of seven samples and probability

Difference between Theoretical and Observed Discrete Probability Distributions

Chart 1 compares the theoretical and observed two category discrete probability distribution of black cards in 20 cards drawn without replacement from a deck of cards.  This clearly shows the bell-shaped curve, the symmetric line chart of theoretical probability distribution and how different the observed distribution and charts are.

Conclusion

The theoretical two-category probability distribution differs from the observed distribution. The observed data could differ from one set to another because of non-uniformity in the condition in which a card is drawn without replacement from a deck of card.