Consider the experiment of drawing two cards without replacement from an ordinary deck of 52 playing cards.
#1. What are the odds in favor of drawing a spade and a heart?
Coin question...
What are the odds in favor of getting at least one head in three successive flips of a coin?
The odds of drawing a spade are 13/52, then drawing the heart would be 13/51. To find the probability of getting both, multiply the individual probabilities.
Below are the possible outcomes of the tosses.
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
"At least one head" can be rephrased as 1, 2, or 3 heads. How many of the 8 possibilities fit?
To calculate the odds in favor of certain events, we first need to understand the total number of possible outcomes and the number of favorable outcomes.
For the experiment of drawing two cards without replacement from a standard deck of 52 playing cards:
1. The total number of possible outcomes is the total number of ways you can choose 2 cards from a deck of 52, which can be calculated using the combination formula C(n, k) = (n!)/(k!(n-k)!). In this case, n = 52 (number of cards in the deck) and k = 2 (number of cards being drawn), so the total number of possible outcomes is C(52, 2) = (52!)/(2!(52-2)!) = 1326.
2. The number of favorable outcomes, in this case, is the number of ways you can draw a spade and a heart. Since there are 13 spades and 13 hearts in the deck, and we are drawing without replacement, the number of favorable outcomes is 13 * 13 = 169.
So, the odds in favor of drawing a spade and a heart are 169 favorable outcomes to 1326 total possible outcomes, which can be simplified to 13:102.
Moving on to the coin question:
To find the odds in favor of getting at least one head in three successive flips of a coin, we can calculate the probability of getting no heads in three flips and subtract it from 1.
1. The total number of possible outcomes when flipping a coin three times would be 2^3 = 8 (each flip has 2 possibilities: heads or tails).
2. The number of favorable outcomes, in this case, is the number of ways you can get no heads (3 tails). Since each flip has 2 possibilities (heads or tails), the total number of ways you can get tails in all three flips is 2^3 = 8.
Therefore, the number of favorable outcomes for getting at least one head is 8 - 1 = 7.
So, the odds in favor of getting at least one head in three successive flips of a coin are 7 favorable outcomes to 8 total possible outcomes, which can be simplified to 7:8.