Two coins are tossed and a tetrahedral die is thrown. The tetrahedral die has 4 faces numbered 1, 2, 3, 4.
(ii)Work out the probabilities for the following outcomes:
(a) only one head and an even number;
(b) no heads and a 4;
(c) a head and a tail (in any order) and an odd number.
outcomes of the two coin tosses
HH
HT
TH
TT
a) prob(only one head and even number) = (2/4)(2/4) = 1/4
b) prob(no heads, and 4) = (1/4)(1/4) = 1/16
c) prob(head and tail and odd) = (2/4)(2/4) = 1/4
(a) Ah, the famous one-head-and-an-even-number combo! It's like finding a unicorn riding a unicycle. To calculate the probability, let's break it down. We have two coins, so the probability of getting only one head is 2/4, which reduces to 1/2. The tetrahedral die has two even numbers out of four, so the probability of getting an even number is 2/4, or 1/2. To find the probability of both events happening together, we multiply them: (1/2) * (1/2) = 1/4. So the probability of getting only one head and an even number is 1/4. Easy peasy lemon squeezy!
(b) Ah, the elusive no-heads-and-a-4 outcome. It's like catching a fish with a donut on the hook! To calculate the probability, we'll use a similar approach. Since there are two coins, the probability of getting no heads is 2/4, or 1/2. The tetrahedral die has only one face numbered 4 out of four, so the probability of getting a 4 is 1/4. To find the probability of both events happening together, we multiply them: (1/2) * (1/4) = 1/8. So the probability of getting no heads and a 4 is 1/8. With odds like that, you might as well go buy a lottery ticket!
(c) Ah, the mysterious head-and-tail and an odd number combo! It's like finding a needle in a haystack shaped like a pretzel. To calculate the probability, let's see what we have. Since there are two coins, there are two possible outcomes: head and tail, or tail and head. Each of these has a probability of 1/2. The tetrahedral die has two odd numbers out of four, so the probability of getting an odd number is 2/4, or 1/2. To find the probability of all events happening together, we multiply them: (1/2) * (1/2) * (1/2) = 1/8. So the probability of getting a head and a tail (in any order) and an odd number is 1/8. Grab your lucky pair of socks and roll the dice!
To find the probabilities for the given outcomes, we will break down each scenario and calculate the probability step by step.
(a) only one head and an even number:
There are three scenarios where we can have only one head and an even number:
1. Head on the first coin, even number on the die.
2. Head on the second coin, even number on the die.
3. Head on the first coin, even number on the die.
Let's calculate the probabilities for each scenario:
1. Head on the first coin, even number on the die:
The probability of getting a head on the first coin is 1/2 (as there are two possible outcomes - head or tail). The probability of getting an even number on the die is 2/4 or 1/2 (since there are two even numbers - 2 and 4, out of four possible outcomes - 1, 2, 3, or 4).
So, the probability of this scenario is (1/2) * (1/2) = 1/4.
2. Head on the second coin, even number on the die:
Similar to the first scenario, the probability of getting a head on the second coin is also 1/2. The probability of getting an even number on the die remains the same, which is 1/2.
So, the probability of this scenario is (1/2) * (1/2) = 1/4.
3. Head on the first coin, even number on the die:
The probability of getting a head on the first coin is 1/2. The probability of getting an even number on the die is 1/2.
So, the probability of this scenario is (1/2) * (1/2) = 1/4.
To find the probability of only one head and an even number, we need to add the probabilities of the three scenarios:
1/4 + 1/4 + 1/4 = 3/4.
Therefore, the probability of getting only one head and an even number is 3/4.
(b) no heads and a 4:
There is only one scenario where we can have no heads and a 4:
1. Tails on both coins, and a 4 on the die.
The probability of getting tails on both coins is (1/2) * (1/2) = 1/4. The probability of getting a 4 on the die is 1/4.
So, the probability of this scenario is (1/4) * (1/4) = 1/16.
Therefore, the probability of getting no heads and a 4 is 1/16.
(c) a head and a tail (in any order) and an odd number:
There are two scenarios where we can have a head and a tail (in any order) and an odd number:
1. Head on the first coin, tail on the second coin, and an odd number on the die.
2. Tail on the first coin, head on the second coin, and an odd number on the die.
Let's calculate the probabilities for each scenario:
1. Head on the first coin, tail on the second coin, and an odd number on the die:
The probability of getting a head on the first coin is 1/2, the probability of getting a tail on the second coin is 1/2, and the probability of getting an odd number on the die is 2/4 or 1/2.
So, the probability of this scenario is (1/2) * (1/2) * (1/2) = 1/8.
2. Tail on the first coin, head on the second coin, and an odd number on the die:
Similar to the first scenario, the probability of getting a tail on the first coin is 1/2, the probability of getting a head on the second coin is 1/2, and the probability of getting an odd number on the die is 1/2.
So, the probability of this scenario is (1/2) * (1/2) * (1/2) = 1/8.
To find the probability of a head and a tail (in any order) and an odd number, we need to add the probabilities of the two scenarios:
1/8 + 1/8 = 2/8 = 1/4.
Therefore, the probability of getting a head and a tail (in any order) and an odd number is 1/4.
In summary:
(a) The probability of only one head and an even number is 3/4.
(b) The probability of no heads and a 4 is 1/16.
(c) The probability of a head and a tail (in any order) and an odd number is 1/4.
To work out the probabilities for the given outcomes, we need to understand the sample space, which is the set of all possible outcomes.
In this scenario, we have two coins and a tetrahedral die. Each coin has two possible outcomes (head or tail), and the die has four possible outcomes (1, 2, 3, or 4).
Therefore, the total number of possible outcomes can be calculated by multiplying the number of outcomes for each component: 2 (coin 1) × 2 (coin 2) × 4 (die) = 16.
Now let's calculate the probabilities for each outcome:
(a) Only one head and an even number:
- We can have a head on either the first or the second coin, which gives us two possibilities.
- The die needs to show an even number, which corresponds to two possible outcomes (2 or 4).
- Therefore, the total number of favorable outcomes is 2 (coin) × 2 (die) = 4.
- The probability is the ratio of favorable outcomes to the total number of outcomes: 4/16 = 1/4.
(b) No heads and a 4:
- Both coins need to land on tails, which is one possible outcome.
- The die needs to show a 4, which is one possible outcome.
- Therefore, the total number of favorable outcomes is 1.
- The probability is 1/16.
(c) A head and a tail (in any order) and an odd number:
- We can have a head on the first coin and a tail on the second or vice versa, which gives us two possibilities.
- The die needs to show an odd number, which corresponds to two possible outcomes (1 or 3).
- Therefore, the total number of favorable outcomes is 2 (coin) × 2 (die) = 4.
- The probability is 4/16 = 1/4.
In summary:
(a) The probability of only one head and an even number is 1/4.
(b) The probability of no heads and a 4 is 1/16.
(c) The probability of a head and a tail (in any order) and an odd number is 1/4.