Coaxial cable balun for a 433 MHz dipole antenna

I recently made a dipole antenna for a frequency of 433 MHz. I took two copper sections, each 2.4 mm in diameter and each 17 cm long. I directly connected a coaxial cable with an impedance of 50 ohms to the dipole.

I installed an RF probe with an LED next to the dipole. It's used to registration the emitted radio signal. At a distance of 20 cm from the antenna, the RF probe's LED doesn't light when I press the transmit button on the radio station. 

Now I've placed the RF probe on the right antenna element, at its very end. The central conductor of the coaxial cable is connected to this element. The RF probe's LED lights up brightly when you press transmit on the radio.

Now I placed the RF probe on the left element of the antenna and the probe LED does not light up.

This antenna is suitable for receiving radio signals, but it can't be used for transmitting.

Why didn't the RF probe LED on the left antenna element light up, and why can't I connect the coaxial cable directly to the dipole? The problem is that the cable is unbalanced, while the antenna is symmetrical. Furthermore, the dipole antenna's impedance is approximately 73 ohms, while the cable's is 50 ohms. Therefore, the radio signal will poorly reach the antenna.

To ensure the radio signal from the radio station reaches the antenna properly, a balun is needed. I made it from a piece of coaxial cable with an impedance of 50 ohms and a length of approximately 22 cm. I didn't do any calculations, but based the cable length on an estimate of one-third of a wavelength at 433 MHz. I don't know how accurate this is, and I wanted to test whether this balun would work.

I connected the balun exactly as shown in the picture.

This is what it looks like now

Now all that's left is to check whether the dipole antenna will perform better. I placed the RF probe 20 cm from the antenna, pressed transmit on the radio, and the LED glowed brightly. This indicates that the radio signal is now stronger with the balun than without it.

Now I've installed an RF probe on the end of the left antenna element, and the LED is glowing brightly. This indicates that the left antenna element is emitting a radio signal, and radio communication with this antenna will be excellent. Now you can not only receive radio signals with this antenna, but also transmit.