Video Transcript
Amelia has these 10 cards. They are six, seven, 10, 15, 16,
19, 22, 29, 32, and 36. Choose a Venn diagram for the
experiment of randomly picking a card that shows two events “picking a multiple of
three” and “picking a square number.” Are the events “picking a multiple
of three” and “picking a square number” mutually exclusive? What is the probability of picking
a number that is a multiple of three and a square number?
We are asked to give this answer as
a fraction in its simplest form. In this question, we are given two
possible events, “picking a multiple of three” and “picking a square number.” The multiples of three are those
numbers in the three times table. They are three, six, nine, 12, 15,
and so on. From the 10 cards Amelia has, six
is a multiple of three. 15 is also a multiple of three. And finally, 36 is a multiple of
three. Out of the 10 cards, three of them
have a multiple of three written on them. This means that the probability of
selecting a card that is a multiple of three is three-tenths.
The square numbers are the numbers
obtained when we multiply a positive integer by itself. One multiplied by one is one, two
multiplied by two is four, three multiplied by three is nine, and so on. The first five square numbers are
one, four, nine, 16, and 25. Out of the numbers on Amelia’s
cards, we have two square numbers, 16 which is four squared and 36 which is six
squared. The probability of picking a card
that has a square number on it is therefore equal to two-tenths.
In terms of the Venn diagrams
given, we need the numbers six, 15, and 36 inside the circle, representing the
multiples of three. This rules out options (A), (B),
and (C). We also need the numbers 16 and 36
inside the circle, representing the square numbers. This rules out option (D). The numbers seven, 10, 19, 22, 29,
and 32 must all lie outside both of the circles. The number 36 is a multiple of
three and a square number. This means that 36 needs to lie in
the intersection of our two circles. The Venn diagram that represents
the experiment of randomly picking a card that shows the two events “picking a
multiple of three” and “picking a square number” is option (E).
The second part of our question
asks us if the two events are mutually exclusive. We know that two events are
mutually exclusive if they cannot happen at the same time. This means that the probability of
their intersection must be zero. However, in this case we have seen
that the number 36 is both a multiple of three and a square number. We can therefore conclude that the
answer is no. “Picking a multiple of three” and
“picking a square number” are not mutually exclusive events.
In the final part of this question,
we are asked to find the probability of picking a number that is a multiple of three
and a square number. This is the probability of picking
a number that is in the intersection of the two events. There is one such number, the
number 36. The probability of picking a number
that is a multiple of three and a square number is one-tenth. This fraction is already written in
its simplest form.
