To solve this problem, we need to use the rules of probability.
A standard deck of 52 cards has 4 different suits: hearts, diamonds, clubs and spades. Each suit has 13 cards, therefore, there are 13 heart cards in a deck.
1. First, we compute the probability of drawing one heart card out of the 52 cards on the first draw. Since there are 13 heart cards, the probability of drawing a heart is 13/52, which simplifies to 0.25 or 25%. This means there's a 25% chance that the first card drawn is a heart.
2. We then draw a second card. However, since we've already drawn one card (assumed to be a heart), there are now 51 cards left in the deck, and only 12 heart cards remain. So, the probability of drawing another heart is 12/51, which approximates to 0.23529411764705882. That's roughly a 23.53% chance of drawing a heart as the second card, given that the first card drawn was a heart.
3. Now, let's draw a third card. We've already taken 2 heart cards from the deck, so there are now 50 cards left, with 11 hearts remaining. The probability of drawing a heart this time is 11/50, which simplifies to 0.22 or about 22%. So, there's a 22% chance of drawing a heart on the third draw.
4. The total probability we're seeking is the outcome in which all three cards drawn are hearts. To find this, we multiply the individual probabilities of each event happening. This is because, in probability, when we want to find the chance of more than one independent event happening together, we multiply their individual probabilities.
Hence, we multiply 0.25 (the probability of drawing a heart first) by 0.23529411764705882 (the probability of drawing a heart second) and then by 0.22 (the probability of drawing a heart third).
This gives us 0.012941176470588235.
So, there is an overall probability of approximately 0.01294 or 1.294% chance that all three cards drawn from a standard deck at random are hearts.