The probability that a person will win a game is `(2)/(3)` and the probability that he will not win a horse race is `(5)/(9)` . If the probability of getting in at least one of the events is `(4)/(5)` ,what is the probability that he will be successful in both the events ?

Answer

To solve the problem step by step, we will denote the events as follows:
- Let event A be the person winning the game.
- Let event B be the person winning the horse race.

### Step 1: Identify the probabilities given in the problem
- The probability that a person will win the game (P(A)) is given as:
\[ P(A) = \frac{2}{3} \]
- The probability that he will not win the horse race (P(B')) is given as:
\[ P(B') = \frac{5}{9} \]

### Step 2: Calculate the probability of winning the horse race (P(B))
Since the probability of not winning the horse race is given, we can find the probability of winning the horse race using the complement rule:
\[ P(B) = 1 - P(B') = 1 - \frac{5}{9} = \frac{4}{9} \]

### Step 3: Identify the probability of at least one event occurring (P(A ∪ B))
The probability of getting at least one of the events (winning the game or winning the horse race) is given as:
\[ P(A \cup B) = \frac{4}{5} \]

### Step 4: Use the formula for the probability of the union of two events
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We can rearrange this formula to find the probability of both events occurring (P(A ∩ B)):
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]

### Step 5: Substitute the known values into the equation
Now we can substitute the values we have:
\[ P(A) = \frac{2}{3}, \quad P(B) = \frac{4}{9}, \quad P(A \cup B) = \frac{4}{5} \]
Substituting these into the equation gives:
\[ P(A \cap B) = \frac{2}{3} + \frac{4}{9} - \frac{4}{5} \]

### Step 6: Find a common denominator and calculate
To perform the addition and subtraction, we need a common denominator. The least common multiple of 3, 9, and 5 is 45. We convert each fraction:
- \(\frac{2}{3} = \frac{30}{45}\)
- \(\frac{4}{9} = \frac{20}{45}\)
- \(\frac{4}{5} = \frac{36}{45}\)

Now substitute these values:
\[ P(A \cap B) = \frac{30}{45} + \frac{20}{45} - \frac{36}{45} \]
\[ P(A \cap B) = \frac{30 + 20 - 36}{45} = \frac{14}{45} \]

### Final Answer
The probability that he will be successful in both events is:
\[ \boxed{\frac{14}{45}} \]