The Masterfoods company says that before the introduction of...

The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&M’s, red another 20%, and orange, blue, and green each made up 10%. The rest were brown. a) If you pick an M&M at random, what was the prob-ability that 1) it is brown? 2) it is yellow or orange? 3) it is not green? 4) it is striped? b) If you pick three M&M’s in a row, what is the prob- ability that 1) they are all brown? 2) the third one is the first one that’s red? 3) none are yellow? 4) at least one is green?

The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&M’s, red another 20%, and orange, blue, and green each made up 10%. The rest were brown.
a) If you pick an M&M at random, what was the prob-ability that
1) it is brown?
2) it is yellow or orange?
3) it is not green?
4) it is striped?
b) If you pick three M&M’s in a row, what is the prob- ability that
1) they are all brown?
2) the third one is the first one that’s red?
3) none are yellow?
4) at least one is green?

Key Concepts

Complement Rule

The complement rule states that the probability of an event not occurring is equal to one minus the probability of the event occurring. This concept is particularly useful when it is easier to calculate the likelihood of the opposite event rather than the event itself.

Multiplication Rule for Independent Events

The multiplication rule is used when determining the probability that several independent events all occur. For independent events, where the outcome of one does not affect the others, the overall probability is the product of the individual event probabilities.

Basic Probability

Basic probability involves measuring the likelihood of an event occurring, usually expressed as the ratio of favorable outcomes to the total number of outcomes in a defined sample space. It is the foundational concept that applies to any random experiment or selection process.

Addition Rule in Probability

The addition rule is used to calculate the probability of the occurrence of one event or another. When the events are mutually exclusive, you can sum their separate probabilities, and when they are not, adjustments are made by subtracting the probability of their intersection to avoid double counting.

Sequential Probability and Conditional Events

Sequential probability deals with events that occur in a specific order, especially when the outcome of one event might affect the conditions of the following events, such as identifying the first occurrence of a particular outcome. This concept often requires applying conditional probability to account for the sequence in which events occur.

Transcript

00:01 Okay, let's start with part 8.

00:07 Okay, for number one, remember that the sum of the probabilities of all possible outcomes in an experiment where we have all is equal to 1.

00:18 And in order to get the probability that it is brown, we'll just have to subtract the sum of the probabilities of the other colors, then subtract it from 1.

00:29 And so we'll have the probability that we get a brown is equal to 1 minus sum 0 .20 plus 0 .20 plus 0 .10 plus 0 .10 and plus 0 .10.

00:55 Which is equal to 1 minus 0 .70 or 0 .2.

01:05 Next.

01:05 Next, for number two, to solve for the probability that it is either a yellow or an orange, we'll use the addition to the group under the mutually exclusive events since the two events cannot exist at the same time of picking.

01:26 So we'll just have probability that it is either yellow or orange is equal to the probability that is yellow plus probability that is an orange which is equal to 0 .20 or 0 .10 equals 0 .30.

02:02 Next for number 3 now that the probability that it is not green means that you already get the probability of the complement green and by the complement rule we have the probability of not green or the probability of the complement of green so we'll put a superscript of c about green is equal to one minus the probability that we get a green that is one minus zero point ten equals zero point ninety and last for part a right remember that the only that only plain m &ms are involved in this experiment and we know that the probability that the event will not, that the event will occur if it is not included in the sample space is just zero.

03:05 Therefore the probability of getting a striped when in fact we only have plain m &m is zero.

03:18 Next for part b we have three eminemes that are picked in sequence.

03:29 So let us put it here.

03:35 So for part b, so for this one we'll use the multiplication with under independent events, that is the probability of these events will occur just equal to the product of the individual probabilities.

03:54 And so for number one, the probability that all of the three are brown okay so the probability that all are brown okay it's just equal to the probability that we get a brown times the probability that we get a brown for this second one and the probability that we get a brown for the third one so we just have to multiply them all we have 0 .30 times 0 .30 times 0 .30 equals 0 .027.

04:51 Next, for number 2, okay...

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