Matthew was at the local dog park one winter afternoon collecting data on the 50 dogs that were there. There were 39 dogs wearing collars. There were seven dogs wearing a coat, but not a collar. There were 31 dogs with a collar, but not wearing a coat. Part a) Complete the two-way table.
The two-way table organizes the dogs according to whether they’re wearing coats, collars, both, or neither. For example, this cell here represents all the dogs who are wearing both coats and collars. This cell represents all the dogs who are wearing collars, but aren’t wearing coats. The last column and the last row give the row and column totals.
Let’s look at the information in the question in order to complete the table. We’re told first of all that the total number of dogs is 50. This is the overall total. So it goes in this cell here.
Next, we’re told that there were 39 dogs who were wearing collars. This is the total number of dogs wearing collars regardless of whether or not they were wearing coats. So it gives the total for the first column, that is the collar column of the table.
We can use this information to work out the number of dogs who weren’t wearing collars. As these two totals must add to the overall total of 50, we can subtract 39 from 50 to give 11. So there’re 11 dogs who weren’t wearing collars.
Next, we’re told that there were seven dogs wearing a coat, but not a collar. So we can fill this in and it’s this cell here. We can use this information to work out the number of dogs who weren’t wearing collars or coats. As the column total is 11, we can subtract seven from 11 to find the value in this cell. There were four dogs wearing neither coats nor collars.
The last piece of information we’re given is that there were 31 dogs with a collar, but not wearing a coat. That gives this cell here. We can then work out the number of dogs wearing both a collar and a coat as the column total should be 39. So subtracting 31 from 39 gives eight.
Now, we filled in all of the main cells in the table, but we need to complete the row totals. Firstly, the total number of dogs who were wearing coats is found by adding eight and seven together to give 15. And then, the total number of dogs who were wearing coats can be found by adding 31 and four together to give 35.
Now that we’ve completed our table, there were a couple of checks that we should do. Firstly, we should check that the sum of our row totals gives the overall total. 15 plus 35 is indeed 50. So this is correct. We already know that the sum of our column totals gives 50 as we use the value of 39 to work out the value of 11.
The other check we should do is to add up the four numbers in the main part of the table and confirm that they do add to 50. Eight plus seven gives 15, adding 31 gives 46, and adding four gives 50. So our overall total is correct. We’ve therefore completed the two-way table.
Part b) One of the dogs is chosen at random, what is the probability that the dog is wearing a collar and a coat?
If the dog is chosen at random, then this just means that each of the individual dogs is equally likely to be chosen. To find the probability that a randomly chosen dog is wearing a collar and a coat, we need to find the number of dogs who are wearing a collar and coat and then divide it by the total. Looking at our table, the number of dogs who are wearing both a collar and a coat is eight. And remember the total is 50. So we have the fraction eight out of 50.
Now, this would actually be enough to get all of the marks on this part of the question. But it is always good practice to simplify a fraction if we can. As both the numerator and the denominator of this fraction are even numbers, we can cancel down by a factor of two. Dividing eight by two gives four and dividing 50 by two gives 25.
The probability that a randomly chosen dog is wearing a collar and a coat is four over 25.