WEBVTT
00:00:02.040 --> 00:00:09.200
Mason spins two four-sided spinners which are both numbered from one to four.
00:00:10.360 --> 00:00:19.440
Let 𝐴 be the event that he spins two numbers with a product of four, and let 𝐵 be the event that he spins two even numbers.
00:00:20.720 --> 00:00:23.440
There are four parts to this question.
00:00:24.400 --> 00:00:28.280
Find the probability of 𝐴, giving your answer as a fraction in its simplest form.
00:00:29.080 --> 00:00:32.800
Find the probability of 𝐵, giving your answer as a fraction in its simplest form.
00:00:33.720 --> 00:00:38.520
Find the probability of 𝐴 intersection 𝐵, giving your answer as a fraction in its simplest form.
00:00:39.280 --> 00:00:45.680
And finally, determine the value of the probability of 𝐴 union 𝐵, giving your answer as a fraction in its simplest form.
00:00:47.640 --> 00:00:53.560
We will begin by clearing some space whilst recalling we need to simplify all of our answers.
00:00:54.560 --> 00:01:03.920
We are told in the question that Mason is spinning two four-sided spinners that are numbered from one to four.
00:01:04.800 --> 00:01:08.760
This means that there is 16 possible combinations.
00:01:09.840 --> 00:01:18.320
These include a one on both spinners all the way up to a four on both spinners.
00:01:19.400 --> 00:01:28.920
We are told that 𝐴 is the event that he spends two numbers with a product of four.
00:01:29.760 --> 00:01:33.120
The word product means multiply or times.
00:01:33.960 --> 00:01:38.040
We know that one multiplied by four is four.
00:01:39.080 --> 00:01:42.000
Two multiplied by two is equal to four.
00:01:42.840 --> 00:01:46.320
And four multiplied by one is also equal to four.
00:01:47.400 --> 00:01:57.960
There are three possible ways that Mason can spin the two spinners such that the numbers have a product of four.
00:01:59.040 --> 00:02:03.120
The probability of 𝐴 is therefore equal to three out of 16 or three sixteenths.
00:02:03.960 --> 00:02:13.000
Since the numerator and denominator have no common factor apart from one, this fraction is in its simplest form.
00:02:14.320 --> 00:02:24.360
We are also told that 𝐵 is the event that Mason spins two even numbers.
00:02:25.400 --> 00:02:36.280
This occurs when he spends a two and a two, a two and a four, a four and a two, or a four and a four.
00:02:37.800 --> 00:02:43.360
There are four possible ways of spinning two even numbers.
00:02:44.360 --> 00:02:48.600
Therefore, the probability of 𝐵 is equal to four sixteenths.
00:02:49.880 --> 00:02:53.720
Both the numerator and denominator here are divisible by four.
00:02:54.360 --> 00:03:00.160
The fraction four sixteenths in its simplest form is therefore one-quarter.
00:03:01.160 --> 00:03:07.880
This is the probability that Mason spends two even numbers.
00:03:09.440 --> 00:03:14.560
The third part of our question asks us to calculate the probability of 𝐴 intersection 𝐵.
00:03:15.440 --> 00:03:18.880
This is the probability that both 𝐴 and 𝐵 occur.
00:03:20.200 --> 00:03:31.360
Mason needs to spin two numbers with a product of four, and they must both be even.
00:03:32.720 --> 00:03:47.360
The only way this is possible is when he rolls two twos, as two multiplied by two is four and the number two is even.
00:03:48.360 --> 00:03:53.680
The probability of 𝐴 intersection 𝐵 is therefore equal to one sixteenth.
00:03:55.320 --> 00:04:00.680
The final part of our question asks us to determine the value of the probability of 𝐴 union 𝐵.
00:04:01.760 --> 00:04:06.800
This is the probability that 𝐴 occurs or 𝐵 occurs or they both occur.
00:04:07.840 --> 00:04:22.040
One way of calculating this is using the addition rule of probability, which states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵.
00:04:23.000 --> 00:04:30.000
In this question, we have already calculated the probability of 𝐴, the probability of 𝐵, and the probability of 𝐴 intersection 𝐵.
00:04:30.960 --> 00:04:41.480
The probability of 𝐴 union 𝐵 is therefore equal to three sixteenths plus four sixteenths minus one sixteenth.
00:04:42.520 --> 00:04:45.280
This is equal to six sixteenths.
00:04:46.360 --> 00:04:51.080
We can check this by looking at the list of possible outcomes.
00:04:52.520 --> 00:05:01.840
Six of the 16 outcomes were in either event 𝐴 or event 𝐵 or both.
00:05:02.960 --> 00:05:12.720
As both six and 16 are divisible by two, we can simplify this fraction to three-eighths.
00:05:13.880 --> 00:05:20.120
The value of the probability of 𝐴 union 𝐵 is three-eighths.
00:05:20.840 --> 00:05:34.560
And the four answers to the question are three sixteenths, one-quarter, one sixteenth, and three-eighths, where all four fractions are written in their simplest form.